Research ArticleCoordination of Supply Chain with One Supplier and TwoCompeting Risk-Averse Retailers under an Option Contract

Rui Wang12 Shiji Song1 and Cheng Wu1

1Department of Automation Tsinghua University Beijing 100084 China2Department of Basic Science Military Transportation University Tianjin 300161 China

Correspondence should be addressed to Shiji Song shijismailtsinghuaeducn

Received 23 October 2015 Accepted 31 December 2015

Academic Editor Paulina Golinska

Copyright copy 2016 Rui Wang et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies an option contract for coordinating a supply chain comprising one risk-neutral supplier and two risk-averseretailers engaged in promotion competition in the selling season For a given option contract in decentralized case each risk-averse retailer decides the optimal order quantity and the promotion policy by maximizing the conditional value-at-risk of profitBased on the retailersrsquo decision the supplier derives the optimal production policy by maximizing expected profit In centralizedcase the optimal decision of the supply chain system is obtained Based on the decentralized and centralized decision we find thecoordination conditions of the supply chain system which can optimize the supply chain system profit and make the profits of thesupply chain members achieve Pareto optimum As for the subchain we also find the coordination conditions which generalizethe results of the supply chain with one supplier and one retailer Our analysis and numerical experiments show that there exists aunique Nash equilibrium between two retailers and the optimal order quantity of each retailer increases (decreases) with its own(competitorrsquos) promotion level

1 Introduction

The current business environment is full of uncertaintiesincluding market demand risk preference lead times pricefluctuation and transport These uncertainties entail newrisks in matching supply and demand [1] and these risksinevitably lead to supply chain inefficiency Such problemshave been encountered by companies such as Mattel Inc atoy maker [2]

As supply chains become more and more complextheir coordination is an increasingly significant challenge forsupply chain agents in industry With channel coordinationsupply chain efficiency has been tremendously improvedand issues involving double marginalization have been wellresolved Coordination among supply chain agents by settingincentive alignment contracts is a hot topic in supply chainmanagement Various contracts have been shown to achievecoordination in supply chains such as buyback contracts[3] quantity-flexibility contracts [4 5] sales-rebate contracts[6] quantity-discount contracts [7ndash9] and revenue-sharingcontracts [10] The option contract is a useful tool to hedge

the risk of operations management and can coordinate thesupply chain effectively [1 2 11ndash15]

In fact option contracts have been widely used inmany industries and are becoming popular in supply chainmanagement For example option contracts are used for35 of Hewlett-Packardrsquos procurement value In particularits purchases of memory chips involve option contracts withits suppliers [11] Boeing offers option contracts to airlines forpurchase of aircraft [16]

Ritchken and Tapiero [17] introduced the option con-tract to hedge against price and quantity fluctuation risksin inventory control Tsay [18] showed that the optioncontract can be used to coordinate the supply chain andincrease buyersrsquo replenishment flexibility Early studies ofthis issue were based on the assumption that supply chainagents were risk-neutral [2 11 13 14 19] and few papersconsidered agentsrsquo risk preference (such as loss aversion)In fact the decision-making behavior of managers hasbeen found to deviate from maximizing (expected) profitin a way consistent with loss aversion [20] as shown bymany experimental studies and observations of managerial

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1970615 11 pageshttpdxdoiorg10115520161970615

2 Mathematical Problems in Engineering

decision-making under uncertainty [21ndash25] To the authorsrsquoknowledge Gan et al [26] were the first to study supplychain coordination issues with loss-averse agents Xu [27]obtained the loss-averse retailerrsquos optimal ordering policyand the supplierrsquos optimal decision under an option contractand demonstrated that both sides can benefit Zhao etal [14] demonstrated that option contracts can coordinatethe supply chain and achieve Pareto improvement using acooperative game approach Zhao et al [15] explored supplychain coordination with bidirectional option contracts andderived a closed-form expression for the retailerrsquos optimalorder strategies with a general demand distribution Chenet al [1] investigated supply chain coordination issues witha risk-neutral supplier and a risk-averse retailer and provedthat option contracts can make the supply chain achieve thePareto optimum Obviously the above research is only con-cerned with the coordination of supply chain upstream anddownstream

In the real environment the supply chain coordinationshould be involved in both the upstream and downstreambut also in the same layer Ingene and Parry [28] exploredthe wholesale pricing behavior within a two-level verticalchannel consisting of amanufacturer selling throughmultipleindependent retailers and analyzed the optimality of channelcoordination for all channel members Padmanabhan andPng [29] examined the manufacturerrsquos return policy for twocompeting retailers with and without demand uncertaintyYao et al [30] further considered the manufacturerrsquos returnpolicy in a newsboy model in which the random demandfaced by two competing retailers was sensitive to retailprice Xiao et al [31] studied a supply chain coordinationmodel with a price-subsidy rate contract in which therewere one manufacturer and two competing retailers Xiaoand Qi [32] investigated the coordination of a supply chainwith one manufacturer and two competing retailers usingtwo mechanisms after the production cost of the manu-facturer was disrupted However above researches did notinvolve option contracts and the supply chain agents arerisk-neutral

Generally high risk implies high return but also highloss A risk-averse individual would prefer to receive arelatively lower return to avoid a potential huge loss Whendecision-makers focus on risk control a number of risk-related performance criteria may be used to measure therisk caused by uncertainty in supply chain managementSpecifically conditional value-at-risk defined as theweightedaverage of value-at-risk and losses strictly exceeding VaR hasattracted much attention in recent years and has been widelyused in finance insurance and supply chain managementCVaR is a consistent risk measure with better properties andcomputing performance than other measures [33 34] Yanget al [35] researched supply chain coordination with a risk-averse retailer and a risk-neutral supplier and showed thatthe supply chain can achieve channel coordination in a CVaRframework with revenue-sharing buyback two-part tariffsand flexible-quantity contracts Chen et al [1] discussedsupply chain coordinationwith a loss-averse retailer under anoption contract although they used the prospective theory todescribe the retailerrsquos risk preference Chen et al [36] studied

stable and coordinating contracts for a supply chain withmultiple risk-averse suppliers under a CVaR objective Wu etal [37] considered a risk-averse newsvendor problem withquantity and price competition under the CVaR criterionHsieh and Lu [38] extended the work of Padmanabhanand Png [29] and Yao et al [30] researched manufacturerrsquosreturn policy in a two-stage supply chain with two risk-averseretailers and random demand via CVaR

Different from the work of Zhao et al [14] Chen et al[36] Wu et al [37] and Hsieh and Lu [38] we investigate thecoordination of supply chain with a risk-neutral supplier andtwo risk-averse retailers engaged in promotional competitionbased on an option contract and a CVaR criterionThis paperwill focus on the following problems (1)What are the optimalorder quantity and the promotion policies of the two risk-averse retailers engaged in promotional competition with aCVaR criterion andwhat is the risk-neutral supplierrsquos optimalproduction decision in the presence of an option contractand a CVaR criterion What are the characteristics of thesedecisions (2) Does there exist a unique equilibrium pointbetween two risk-averse retailers engaged in promotionalcompetition (3)How can a supply chainwith two risk-averseretailers engaged in promotional competition be coordinatedbased on an option contract and a CVaR criterion (4) Whatis the impact of the promotional competition on the retailerrsquosoptimal ordering policy the supplierrsquos optimal productiondecision and supply chain coordination

To address these problems the following scenario willbe studied There are two risk-averse retailers engaged inpromotional competition who are selling the same productsThe demand for each retailer is stochastic and dependspartially on the retailerrsquos promotion level In the decentralizedcase the objective of each risk-averse retailer is to maximizehis conditional value-at-risk of profit and the supplierrsquosgoal is to maximize his expected profit The existence ofequilibrium point between two risk-averse retailers engagedin promotional competition will then be explored In thecentralized case the optimal decisions for the supply chainsystem-wide will be discussedThe purpose of this study is toinvestigate supply chain coordination issues and to explorethe impact of promotional competition on the retailersrsquooptimal order quantity the supplierrsquos optimal productiondecision and supply chain coordination To the best of theauthorsrsquo knowledge this problem has not been consideredin a situation with an option contract and a CVaR criterionThis paper differs from existing research on supply chaincoordination in three aspects First there exists a uniqueNash equilibrium between two retailers and the impacts ofthe promotion level on the retailerrsquos equilibrium option orderquantity are analyzed Secondly the impact of a shortagepenalty cost on the supplierrsquos optimal production decisionis considered Finally the coordination conditions for sucha supply chain under the option contract and the CVaRcriterion are obtained

The remainder of this paper is organized as followsSection 2 presents the model formulation and the assump-tions made Section 3 describes the decentralized caseinvolving the optimal option order quantity and the promo-tion policy of two retailers under an option contract and

Mathematical Problems in Engineering 3

a CVaR criterion and the supplierrsquos optimal production deci-sion under an option contract The equilibrium competitionbetween two risk-averse retailers is also analyzed Section 4explores supply chain system and its subchain coordinationissues and the corresponding coordination conditions aregiven Section 5 illustrates the impact of the promotionlevel on the optimal order quantity of each retailer throughnumerical experiments Section 6 concludes the paper

2 Model Formulation and Assumptions

Let us consider a one-period two-echelon supply chaincoordination problemThe supply chain consists of one risk-neutral supplier and two risk-averse retailers engaged inpromotion competition In the traditional Cournot compe-tition the demand is considered as the determination [10]but in fact the demand is uncertain We assume that tworetailers order the product from the supplier with demanduncertainty The uncertain demand faced by retailer 119894 (119894 =1 2) is 119863

119894 which takes on an additive form and can be

expressed as 119863119894= 119889119894+ 119883119894 119889119894⩾ 0 is a demand relating

to the market scale and promotion level We assume that119889119894= 119889119894(119890119894 1198903minus119894) is a linear function increasing (decreasing)

monotonically with the retailerrsquos (the competitorrsquos) promo-tion level 119890

119894(1198903minus119894) which can be described by the amount of

promotional products Let 119892119894(119890119894) denote the promotional cost

of retailer 119894 which is a second-order differentiable functionsatisfying 119892

119894(0) = 0 Let 119866

119894= 1198921015840

119894(119890119894) be a monotonically

increasing function with its inverse function denoted as119866minus1119894

119867119894= 11989210158401015840

119894(119890119894) gt 0 119883

119894is a continuous differentiable and

invertible random variable which is independent of 119889119894 For

simplicity let 119883119894have the same probability density function

119891(119909) and cumulative distribution function 119865(119909) 119865(119909) isnonnegative strictly increasing and invertible and satisfies119865(0) = 0 and 119865(119909) = 1 minus 119865(119909) 119863 is total market demand119863 = 119863

1+ 1198632 which increases monotonically with the total

promotion levelThe product is perishable with a comparatively long

lead time and a short selling season This paper focuses onactivities from the beginning of the production season to theend of the selling season At the beginning of the productionseason retailer 119894 and the supplier sign an option contract withtwo parameters denoted as (119874

119894 119864119894) where 119874

119894is the option

price and 119864119894is the exercise price At the same time each

retailer purchases an option quantity denoted as 119902119894 at unit

price 119874119894 Then the supplier makes his production decision

based on the retailersrsquo option order quantities and begins toproduce In the selling season depending on actual demandretailers begin to exercise their option quantitiesTheunit saleprice of retailer 119894 is 119875

119894 and the unit production cost of the

supplier is119862 If the supplier fails to complete the option orderquantity exercised by the retailers then the retailers have theright to punish the supplier by shortage penalty cost denotedas119885119894 which is the cost to the supplier to obtain an additional

unit of product by expediting production or buying from analternative source

Without loss of generality it is assumed that the salvage ofthe supplier and retailers is zero and that there are no credit

losses for retailers in an out-of-stock situation To avoid trivialproblem and to ensure profit for all parties it is assumed that119862 lt 119874

119894+ 119864119894lt 119875119894 The notation 119909+ = max0 119909 will be used

3 Retailersrsquo Optimal Option Order Policy

At the beginning of the production season retailer 119894 pur-chases an option order quantity 119902

119894with an option contract

When retailer 119894 is risk-neutral the profit of retailer 119894 denotedas 120587119903119894(119902119894 119890119894 119863119894) is

120587119903

119894(119902119894 119890119894 119863119894) = 119875119894min (119902

119894 119863119894) minus 119874119894119902119894minus 119864119894min (119902

119894 119863119894)

minus 119892119894(119890119894)

(1)

In (1) the first term is the sales revenue the second term isthe option cost the third term is the exercise cost and thelast term is the promotional cost

Then the corresponding expected profit denoted as119864[120587119903

119894] is

119864 [120587119903

119894] = (119875

119894minus 119874119894minus 119864119894) 119902119894minus 119892119894(119890119894)

minus (119875119894minus 119864119894) int

119902119894minus119889119894

0

119865 (119909) 119889119909

(2)

Because retailer 119894 is risk-averse the degree of risk aversionshould be taken into account in determining the optionorder quantity In this paper retailer 119894 takes the CVaR as hisperformance measure because the CVaR risk measure is arelatively conservative decision-making criterion [35]

According to the definition given by Rockafellar andUryasev [33] the definition of the CVaR on retailer 119894rsquos optionorder quantity and promotion level is given by the following

Definition 1 120578119894-CVaR on retailer 119894rsquos option order quantity 119902

119894

and promotion level 119890119894 denoted as 119862

120578119894[120587119903

119894] is

119862120578119894[120587119903

119894] = 119864 120587

119903

119894(119902119894 119890119894 119863119894) | 120587119903

119894(119902119894 119890119894 119863119894)

⩽ 120573120578119894[120587119903

119894(119902119894 119890119894 119863119894)]

(3)

where 120573120578119894[120587119903

119894(119902119894 119890119894 119863119894)] = inf120573 | 119875[120587

119903

119894(119902119894 119890119894 119863119894) ⩽ 120573] ⩾ 120578

119894

is a quantile and 120578119894isin (0 1] reflects the degree of risk aversion

for retailer 119894 (the smaller 120578119894is the more risk-averse retailer 119894

is)

To facilitate the calculation an equivalent definition isgiven by [33 34]

119862120578119894[120587119903

119894] = max120572isinR

120572 minus1

120578119894

119864 [120572 minus 120587119903

119894(119902119894 119890119894 119863119894)]+ (4)

where 120572 is a real number Let

119880119894= 120572 minus

1

120578119894

119864 [120572 minus 120587119903

119894(119902119894 119890119894 119863119894)]+ (5)

Combining expressions (2) and (5) with some algebra yields

4 Mathematical Problems in Engineering

119880119894=

120572 120572 ⩽ 119882119894

120572 minus1

120578119894

int

(120572minus119882119894)119860119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 119882

119894lt 120572 ⩽ 119881

119894

120572 minus1

120578119894

int

119862119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 minus

1

120578119894

int

infin

119862119894

(120572 minus 119881119894) 119891 (119909) 119889119909 120572 gt 119881

119894

(6)

where 119860119894= 119875119894minus 119864119894 119861119894= 119875119894minus 119864119894minus 119874119894 119862119894= 119902119894minus 119889119894 119881119894=

(119875119894minus 119864119894minus 119874119894)119902119894minus 119892119894(119890119894) and119882

119894= (119875119894minus 119864119894)119889119894minus 119874119894119902119894minus 119892119894(119890119894)

Property 1 119862120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

The objective of retailer 119894 is to maximize his CVaRmeasure 119862

120578119894[120587119903

119894] The optimal solution of (3) denoted as

(119902lowast

119894 119890lowast

119894) can be obtained using (3) (6) and Property 1

Theorem 2 Given the degree of risk aversion 120578119894for retailer 119894

and the competitorrsquos promotion level 1198903minus119894

retailer 119894rsquos optimaloption ordering policy is given by

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(7)

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] =

119860119894

120578119894

int

119866120578119894

0

119865minus1(119905) 119889119905 + 119861119894119889119894 (119890

lowast

119894 1198903minus119894)

minus 119892119894(119890lowast

119894)

(8)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Theorem 2 shows that retailer 119894rsquos optimal option orderquantity contains two parts one is related to the promotionalcompetition and the other is determined by stochasticdemand This result is consistent with the form of demandfaced by retailers When retailer 119894 achieves the optimal levelof promotional activity 119890lowast

119894 retailer 119894rsquos option order quantity 119902lowast

119894

satisfies the following property

Property 2 Given the optimal promotion level 119890lowast119894for retailer

119894 then retailer 119894rsquos optimal option order quantity 119902lowast119894satisfies the

following properties 120597119902lowast119894120597119874119894lt 0 120597119902lowast

119894120597119864119894lt 0 120597119902lowast

119894120597120578119894gt 0

and 120597119902lowast1198941205971198903minus119894

lt 0

In Property 2 the first two items show that when theoption price and the exercise price rise retailer 119894 will reducehis cost by reducing the option order quantity to ensure hisprofitThe third item illustrates that the higher 120578

119894is the lower

retailer 119894rsquos degree of risk aversion will be making 119902lowast119894higher

This result is consistent with the intuition that risk-averseretailers would rather have a steady income than take a riskto obtain more benefit When 120578

119894= 1 retailer 119894 is risk-neutral

and retailer 119894rsquos optimal order policy (1199020lowast119894 1198900lowast

119894) is

1199020lowast

119894= 119889119894(1198900lowast

119894 1198903minus119894) + 119865minus1(1198660)

1198900lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(9)

where 1198660= 119861119894119860119894 Obviously 119902lowast

119894lt 1199020lowast

119894and 119890lowast119894lt 1198900lowast

119894 Wang

and Webster [12] derived a similar conclusion howevertheir result is based on a whole-price contract for a supplychain with a loss-averse retailer The last item in Property 2indicates that the higher the competitorrsquos promotion level isthe lower the retailer 119894rsquos optimal option order quantity willbe and corresponding profit will be reduced When retailer1 is in his best promotional environment the upper boundon 119902lowast1is obtained when 119890

2= 0 in (7) and the lower bound

on 119902lowast1is obtained when 119890

2rarr infin in (7) Therefore the curve

1199021= 119902lowast

1(119890lowast

1 1198902) is the reaction curve on (119890

1 1198902) for retailer 1

as illustrated in Figure 1If 119902lowast2(1198901 119890lowast

2) is retailer 2rsquos optimal option order quantity

similar conclusions will be obtained Figure 1 also illustratesthe reaction curve of retailer 2 that is the curve 119902

2=

119902lowast

2(1198901 119890lowast

2) As illustrated in Figure 1 there is an equilibrium

point between the two retailers At the equilibrium pointif each retailer knows the best level of promotional activityof the other then the two retailersrsquo competition satisfies thefollowing equilibrium equations

1198621205781[120587119903

1(119902lowast

1 elowast)] ⩾ 119862

1205781[120587119903

1(1199021 1198901 119890lowast

2)] forall119902

1 1198901⩾ 0

1198621205782[120587119903

2(119902lowast

2 elowast)] ⩾ 119862

1205782[120587119903

2(1199022 1198902 119890lowast

1)] forall119902

2 1198902⩾ 0

(10)

where elowast = (119890lowast1 119890lowast

2)

The next theorem shows the existence and uniqueness ofNash equilibrium between the two retailers

Theorem 3 Given that each of the two risk-averse retailersknows the best promotion level for the other then there existsa unique Nash equilibrium between the order quantities oftwo retailers engaging in promotion competition and theequilibrium satisfies the following conditions

119902lowast

119894= 119889lowast

119894+ 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(11)

where 119889lowast119894= 119889119894(119890lowast

119894 119890lowast

3minus119894)

Theorem 3 indicates that at the equilibrium point 119902lowast1=

119902lowast

2 it is possible to obtain the relationship of the two retailers

on sale price option price exercise price and degree of riskaversion for retailer 119894 that is119861

1= 1198612and 12057811205782= 11986011198602The

first item indicates that the two retailers have the same profitper unit of product (promotional cost is not considered) andthe second item shows that the two retailersrsquo degree of riskaversion is proportional to the difference between the saleprice and the exercise price (excluding the sale price option

Mathematical Problems in Engineering 5

e1

e2

qlowast

2(e

lowast

2 e1)

qlowast

1(e

lowast

1 e2)

Figure 1 Impact of the promotion level on equilibrium orderquantity

price and exercise price and degree of risk aversion of thetwo retailers is the same)

4 Supplierrsquos Optimal Production Decision

Before the production season the supplier will determinethe production quantity in accordance with the retailersrsquooption order quantity Considering that the two retailers willnot exercise all their options at the beginning of the sellingseason the supplier will reduce production by running therisk of being punished The supplierrsquos profit denoted by 120587119898is

120587119898=

2

sum

119894=1

120587119898

119894=

2

sum

119894=1

119874119894119902lowast

119894+ 119864119894min (119902lowast

119894 119863119894) minus 119862119902

119898

119894

minus 119885119894[min (119902lowast

119894 119863119894) minus 119902119898

119894]+

(12)

where 120587119898

119894is the profit from the supplierrsquos selling of the

production to retailer 119894 and 119902119898

119894is the supplierrsquos option

production for retailer 119894 On the right-hand side of the sumin (12) the first term is the option cost that retailer 119894 paysto the supplier the second term is the supplierrsquos revenuewhen retailers exercise their options the third term is theproduction cost and the last term is the shortage penalty costThe corresponding expected profit is

119864 [120587119898] =

2

sum

119894=1

119864 [120587119898

119894] =

2

sum

119894=1

[(119874119894+ 119864119894) 119902lowast

119894minus 119862119902119898

119894

minus 119864119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

minus 119885119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909]

(13)

Property 3 119864[120587119898] is a second-order differentiable function in119902119898

119894and satisfies the following properties

120597119864 [120587119898

119894]

120597119874119894

gt 0

120597119864 [120587119898

119894]

120597119864119894

gt 0

120597119864 [120587119898

119894]

120597119862lt 0

120597119864 [120587119898

119894]

120597119885119894

lt 0

(14)

Property 3 shows that the higher the option price and theexercise price are the higher 119864[120587119898

119894] will be and the higher

the production cost and the penalty cost are the lower 119864[120587119898119894]

will beThese results are consistent with our intuition and theactual situation

In the proof of Property 3 letting the first-order partialderivative of 119864[120587119898

119894] with respect to 119902119898

119894be equal to zero

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 (119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)) = 0 (15)

According to (15) the supplierrsquos production decision forretailer 119894 denoted as 119902119898Δ

119894 is given by

119902119898Δ

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866119898

119894) (16)

where 119866119898119894= (119885119894minus 119862)119885

119894

The first-order partial derivative of 119902119898Δ119894

with respect to119885119894

is 120597119902119898Δ119894120597119885119894= 119862119885

2

119894119891(119902119898Δ

119894minus 119889119894(119890lowast

119894 1198903minus119894)) gt 0 which implies

that for given 119862 119890lowast119894 and 119890

3minus119894 the supplierrsquos production is

increasing in 119885119894 In other words the supplier will choose to

produce more options to reduce the loss when the shortagepenalty cost increases But if the shortage penalty cost isgreater than a certain critical value the supplier will produceall the options or his marginal loss will be greater than themarginal profit Combining (7) and (15) yields the shortagepenalty threshold value

119885119894=

119860119894119862

119860119894minus 119861119894120578119894

(17)

The first-order partial derivative of 119885119894with respect to 120578

119894is

120597119885119894120597120578119894= 119860119894119861119894119862(119860119894minus 119861119894120578119894)2gt 0 which indicates that

the penalty threshold value 119885119894will be higher when retailer

119894 is not very risk-averse According to the above analysisit is clear that the penalty threshold value directly impactsthe supplierrsquos production decision The supplierrsquos optimalproduction decision for retailer 119894 is given by

119902119898lowast

119894=

119902119898Δ

119894 119885119894lt 119885119894

119902lowast

119894 119885

119894⩾ 119885119894

(18)

From (18) it is known thatwhen119885119894is less than119885

119894 the supplier

will accept the penalty and will produce only 119902119894Δ119898

units of

6 Mathematical Problems in Engineering

options for retailer 119894 otherwise he will produce all optionorder quantity At the same time the total optimal productionquantity 119902119898lowast of the supplier for two retailers is given by

119902119898lowast

=

119902119898Δ

1+ 119902119898Δ

2 1198851lt 1198851 1198852lt 1198852

119902119898Δ

1+ 119902lowast

2 119885

1lt 1198851 1198852⩾ 1198852

119902lowast

1+ 119902119898Δ

2 119885

1⩾ 1198851 1198852lt 1198852

119902lowast

1+ 119902lowast

2 119885

1⩾ 1198851 1198852⩾ 1198852

(19)

When 120578119894= 1 the penalty threshold value is119885

119894= 119860119894119862119874119894

and the corresponding optimal production decision 1199021198980lowast119894

ofthe supplier for retailer 119894 is

1199021198980lowast

119894=

1199021198980Δ

119894 119885119894lt 119885119894

1199020lowast

119894 119885

119894⩾ 119885119894

(20)

Obviously 119885119894lt 119885119894and 119902119898lowast119894

lt 1199021198980lowast

119894

5 Supply Chain Coordination

It is well known that the optimal decision of supply chainsystem-wide is the benchmark for supply chain coordinationTo derive the optimal decision of the channel the supplychain is taken as one entity and the profit of the supply chainsystem is formulated and denoted as 120587119904

120587119904=

2

sum

119894=1

[119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)] (21)

where 119902119904119894and 119890119904119894are respectively the order quantity and the

promotion level for retailer 119894 in the supply chain system Onthe right-hand side of the sum in (21) the first term is thesales revenue the second term is the production cost and thelast term is the promotion cost The corresponding expectedprofit is

119864 [120587119904] =

2

sum

119894=1

119864 [119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)]

=

2

sum

119894=1

[minus119875119894int

119902119904

119894minus119889119904

119894

0

119865 (119909) 119889119909 + (119875119894 minus 119862) 119902119904

119894minus 119892119894(119890119904

119894)]

(22)

Obviously the optimal decision of the supply chain canbe obtained by maximizing (22) Let the first-order partialderivative of 119864[120587119904] with respect to 119902119904

119894and 119890119904119894be equal to zero

120597119864 [120587119904]

120597119902119904

119894

= 119861119904

119894minus 119875119894119865 (119862119904

119894) = 0

120597119864 [120587119904]

120597119890119904

119894

= minus119866119894(119890119904

119894) + 119875119894119865 (119862119904

119894)119898119904

119894119894

+ 1198753minus119894119865 (119862119904

3minus119894)119898119904

3minus119894119894= 0

(23)

where 119862119904119894= 119902119904

119894minus 119889119904

119894 119894 = 1 2 In addition the leading

principleminors ofmatrix of119864[120587119904] are as followsminus119875119894119891(119862119904

119894) lt

0 119875119894119891(119862119904

119894)119867119894+ 1198753minus119894119891(119862119904

119894)119891(119862119904

3minus119894)(1198983minus119894119894

)2gt 0 The optimal

decision for the channel in (22) can be obtained as follows

119902119904lowast

119894= 119889119904lowast

119894+ 119865minus1(119866119904

119894)

119890119904lowast

119894= 119866minus1

119894(119898119904

119894119894119861119904

119894+ 119898119904

3minus119894119894119861119904

3minus119894)

(24)

where 119889119904lowast119894= 119889119894(119890119904lowast

119894 119890119904lowast

3minus119894) 119861119904119894= 119875119894minus 119862 119861119904

3minus119894= 1198753minus119894

minus 119862 119866119904119894=

119861119904

119894119875119894119898119904119894119894= 1198891015840

119894119890119904

119894

and1198981199043minus119894119894

= 1198891015840

3minus119894119890119904

119894

The optimal order quantity and the promotion level of the

supply chain system-wide can be used as the benchmark forsupply chain coordination to adjust the option parametersThe coordination conditions of supply chain will be given bythe following theorem

Theorem 4 The supply chain of a risk-neutral supplier andtwo risk-averse retailers engaging in promotion competitionwith an option contract and a CVaR criterion can be coordi-nated by the following conditions

1198611= 1198612

1205781

1205782

=1198601

1198602

1 minus119862

119875119894

lt 120578119894⩽ 1

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119885119894⩾ 119875119894

119862 lt 119874119894+ 119864119894lt 119875119894

(25)

In Theorem 4 the first two items show that if the supplychain can be coordinated then the two retailers first achievecompetitive equilibriumThe third item implies that retailersare not very risk-averse that is retailers will pursue profit bytaking some risk LetΔ119889

119894= 119889119894(119890119894+Δ119890119894 1198903minus119894)minus119889119894(119890119894 1198903minus119894)Δ119889119904119894=

119889119894(119890119904

119894+Δ119890119894 119890119904

3minus119894)minus119889119894(119890119904

119894 119890119904

3minus119894) andΔ119889119904

3minus119894= 1198893minus119894(119890119904

3minus119894 119890119904

119894+Δ119890119894)minus

1198893minus119894(119890119904

3minus119894 119890119904

119894) where Δ119890

119894is a small change on the promotion

level of retailer 119894 The fourth item shows that if the promotionlevel of retailer 119894 changes Δ119890

119894units then retailer 119894rsquos demand

will change Δ119889119904119894units and the other retailerrsquos demand will

change Δ1198891199043minus119894

units in the centralized case retailer 119894rsquos demandwill change (119861119904

119894Δ119889119904

119894+119861119904

3minus119894Δ119889119904

3minus119894)119861119894units in the decentralized

case This relationship is brought about by the two retailersrsquocompetition and promotional activities Furthermore thepenalty threshold value must be higher than the sale priceor the supplier will not produce all option order quantityFinally by adjusting the parameters of the option contractthe whole supply chain profit can reach the optimum andthe profits of supply chain members can achieve Paretooptimum

Mathematical Problems in Engineering 7

Table 1 Comparison of the coordination conditions of several supply chains

SC1

SC2

119878119862119894

1198781198623119894

119885 ⩾ 119875 119885 ⩾ 119875 119885119894⩾ 119875119894

119885119894⩾ 119875119894

119875 = 119864 +119875119874

119862119875 = 119864 +

119875119874120578

119862 minus 119875(1 minus 120578)119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119875119894= 119864119894+119875119894119874119894

119862

120578 gt 1 minus119862

119875120578119894gt 1 minus

119862

119875119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

S

S1

S2

R1

PromotionCompetitionR2

C1

C2

SC1

SC

SC2

Figure 2 The relationship of 119878119862 agents

Then we will discuss the subchain coordination condi-tions Figure 2 shows that the supply chain contains twosubchains and the coordination conditions of subchain willbe given by the following theorem

Theorem 5 The subchain (119878119862119894) can be coordinated by the

following conditions

120578119894gt 1 minus

119862

119875119894

119885119894⩾ 119875119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

gt 1

(26)

FromTheorem 5 we can see that the first three coordina-tion conditions of 119878119862

119894are similar to that of the entity supply

chain The last condition implies that Δ119889119894gt Δ119889

119904

119894 which

means that the level of the promotion activity of retailer 119894 indecentralized decision is higher than in centralized decision

Furthermore we can derive the coordination conditionsof supply chain with a neutral supplier and a neutral retailer(1198781198621) supply chain with a neutral supplier and a risk retailer

(1198781198622) and subchain of supply chain with a neutral supplier

and two neutral retailers engaging in promotion competition(1198781198623119894) In the coordination conditions of 119878119862

119894 when 120578

119894and 119890119894

are 1 and 0 respectively we get the coordination conditionsof 1198781198623119894and 119878119862

2 which are consistent with the conclusion of

[39] and when 120578119894= 1 and 119890

119894= 0 the coordination conditions

of 1198781198621are obtained Now we compare them in Table 1 From

Table 1 we find that the penalty cost is higher than theretail price for four supply chains or subchains coordination

conditions Obviously this condition is beneficial for theretailers to fully exercise the option and the supplier hasto produce all option order quantity to reduce loss Table 1shows that the sale price of 119878119862

1is higher than 119878119862

2 which

embodies the characteristic of the retailerrsquos risk aversionSimilar conclusions exist in 119878119862

119894and 119878119862

3119894

6 Numerical Analysis

In this section we carry out numerical experiments underthe model assumption to illustrate our findings We let 119875

1=

55 1198641= 335 119874

1= 58 119875

2= 547 119864

2= 326 119874

2=

67 1205781= 068 120578

2= 071 119862 = 275 and 119885

1= 60

and 1198852= 56 For simplicity we assume that the random

demand variable of each retailer is uniformly distributedon [0 300] and 119889(119890

119894 1198903minus119894) = 100 + 5119890

119894minus 043 sdot 5119890

3minus119894 The

retailersrsquo optimal option order policy and supplierrsquos optimalproduction decision in decentralized case and the optimalsupply chain decision system-wide are shown in Table 2 (notethat the data are rounded) In Table 2 retailersrsquo optimal orderquantity is the same as the supplierrsquos production decisionand the level of two retailersrsquo promotional activity remainsconsistent in decentralized case Furthermore the optimalprofit of supply chain system is the same as that of theretailers and the supplier in decentralized case which impliesthat the supply chain consisting of a risk-neutral supplierand two risk-averse retailers in competition and engagedin promotion is coordinated under the option contract andCVaR criterion

Thenwe analyze the impact of the level of the promotionalactivity on the retailersrsquo order quantities by fixing 119890lowast

1= 85

and 119890lowast2= 85 respectively and varying 119890

2and 1198901from 0 to

200 in steps of 5 corresponding 119890lowast1and 119890lowast2 Figure 3(a) shows

that retailer 1rsquos order quantity will increase when 1198901increases

however retailer 2rsquos order quantity will decrease similar toFigure 3(b) Figures 3(a) and 3(b) also illustrate the uniqueequilibrium point between two risk-averse retailers which isin agreement with the conclusion of Theorem 3

7 Conclusion

This paper investigates an option contract for coordinatinga supply chain with one risk-neutral supplier and two risk-averse retailers engaged in promotion competition Based onthe option contract the optimal option order quantity andthe promotion level of two retailers are obtained with CVaR

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

decision-making under uncertainty [21ndash25] To the authorsrsquoknowledge Gan et al [26] were the first to study supplychain coordination issues with loss-averse agents Xu [27]obtained the loss-averse retailerrsquos optimal ordering policyand the supplierrsquos optimal decision under an option contractand demonstrated that both sides can benefit Zhao etal [14] demonstrated that option contracts can coordinatethe supply chain and achieve Pareto improvement using acooperative game approach Zhao et al [15] explored supplychain coordination with bidirectional option contracts andderived a closed-form expression for the retailerrsquos optimalorder strategies with a general demand distribution Chenet al [1] investigated supply chain coordination issues witha risk-neutral supplier and a risk-averse retailer and provedthat option contracts can make the supply chain achieve thePareto optimum Obviously the above research is only con-cerned with the coordination of supply chain upstream anddownstream

In the real environment the supply chain coordinationshould be involved in both the upstream and downstreambut also in the same layer Ingene and Parry [28] exploredthe wholesale pricing behavior within a two-level verticalchannel consisting of amanufacturer selling throughmultipleindependent retailers and analyzed the optimality of channelcoordination for all channel members Padmanabhan andPng [29] examined the manufacturerrsquos return policy for twocompeting retailers with and without demand uncertaintyYao et al [30] further considered the manufacturerrsquos returnpolicy in a newsboy model in which the random demandfaced by two competing retailers was sensitive to retailprice Xiao et al [31] studied a supply chain coordinationmodel with a price-subsidy rate contract in which therewere one manufacturer and two competing retailers Xiaoand Qi [32] investigated the coordination of a supply chainwith one manufacturer and two competing retailers usingtwo mechanisms after the production cost of the manu-facturer was disrupted However above researches did notinvolve option contracts and the supply chain agents arerisk-neutral

Generally high risk implies high return but also highloss A risk-averse individual would prefer to receive arelatively lower return to avoid a potential huge loss Whendecision-makers focus on risk control a number of risk-related performance criteria may be used to measure therisk caused by uncertainty in supply chain managementSpecifically conditional value-at-risk defined as theweightedaverage of value-at-risk and losses strictly exceeding VaR hasattracted much attention in recent years and has been widelyused in finance insurance and supply chain managementCVaR is a consistent risk measure with better properties andcomputing performance than other measures [33 34] Yanget al [35] researched supply chain coordination with a risk-averse retailer and a risk-neutral supplier and showed thatthe supply chain can achieve channel coordination in a CVaRframework with revenue-sharing buyback two-part tariffsand flexible-quantity contracts Chen et al [1] discussedsupply chain coordinationwith a loss-averse retailer under anoption contract although they used the prospective theory todescribe the retailerrsquos risk preference Chen et al [36] studied

stable and coordinating contracts for a supply chain withmultiple risk-averse suppliers under a CVaR objective Wu etal [37] considered a risk-averse newsvendor problem withquantity and price competition under the CVaR criterionHsieh and Lu [38] extended the work of Padmanabhanand Png [29] and Yao et al [30] researched manufacturerrsquosreturn policy in a two-stage supply chain with two risk-averseretailers and random demand via CVaR

Different from the work of Zhao et al [14] Chen et al[36] Wu et al [37] and Hsieh and Lu [38] we investigate thecoordination of supply chain with a risk-neutral supplier andtwo risk-averse retailers engaged in promotional competitionbased on an option contract and a CVaR criterionThis paperwill focus on the following problems (1)What are the optimalorder quantity and the promotion policies of the two risk-averse retailers engaged in promotional competition with aCVaR criterion andwhat is the risk-neutral supplierrsquos optimalproduction decision in the presence of an option contractand a CVaR criterion What are the characteristics of thesedecisions (2) Does there exist a unique equilibrium pointbetween two risk-averse retailers engaged in promotionalcompetition (3)How can a supply chainwith two risk-averseretailers engaged in promotional competition be coordinatedbased on an option contract and a CVaR criterion (4) Whatis the impact of the promotional competition on the retailerrsquosoptimal ordering policy the supplierrsquos optimal productiondecision and supply chain coordination

To address these problems the following scenario willbe studied There are two risk-averse retailers engaged inpromotional competition who are selling the same productsThe demand for each retailer is stochastic and dependspartially on the retailerrsquos promotion level In the decentralizedcase the objective of each risk-averse retailer is to maximizehis conditional value-at-risk of profit and the supplierrsquosgoal is to maximize his expected profit The existence ofequilibrium point between two risk-averse retailers engagedin promotional competition will then be explored In thecentralized case the optimal decisions for the supply chainsystem-wide will be discussedThe purpose of this study is toinvestigate supply chain coordination issues and to explorethe impact of promotional competition on the retailersrsquooptimal order quantity the supplierrsquos optimal productiondecision and supply chain coordination To the best of theauthorsrsquo knowledge this problem has not been consideredin a situation with an option contract and a CVaR criterionThis paper differs from existing research on supply chaincoordination in three aspects First there exists a uniqueNash equilibrium between two retailers and the impacts ofthe promotion level on the retailerrsquos equilibrium option orderquantity are analyzed Secondly the impact of a shortagepenalty cost on the supplierrsquos optimal production decisionis considered Finally the coordination conditions for sucha supply chain under the option contract and the CVaRcriterion are obtained

The remainder of this paper is organized as followsSection 2 presents the model formulation and the assump-tions made Section 3 describes the decentralized caseinvolving the optimal option order quantity and the promo-tion policy of two retailers under an option contract and

Mathematical Problems in Engineering 3

a CVaR criterion and the supplierrsquos optimal production deci-sion under an option contract The equilibrium competitionbetween two risk-averse retailers is also analyzed Section 4explores supply chain system and its subchain coordinationissues and the corresponding coordination conditions aregiven Section 5 illustrates the impact of the promotionlevel on the optimal order quantity of each retailer throughnumerical experiments Section 6 concludes the paper

2 Model Formulation and Assumptions

Let us consider a one-period two-echelon supply chaincoordination problemThe supply chain consists of one risk-neutral supplier and two risk-averse retailers engaged inpromotion competition In the traditional Cournot compe-tition the demand is considered as the determination [10]but in fact the demand is uncertain We assume that tworetailers order the product from the supplier with demanduncertainty The uncertain demand faced by retailer 119894 (119894 =1 2) is 119863

119894 which takes on an additive form and can be

expressed as 119863119894= 119889119894+ 119883119894 119889119894⩾ 0 is a demand relating

to the market scale and promotion level We assume that119889119894= 119889119894(119890119894 1198903minus119894) is a linear function increasing (decreasing)

monotonically with the retailerrsquos (the competitorrsquos) promo-tion level 119890

119894(1198903minus119894) which can be described by the amount of

promotional products Let 119892119894(119890119894) denote the promotional cost

of retailer 119894 which is a second-order differentiable functionsatisfying 119892

119894(0) = 0 Let 119866

119894= 1198921015840

119894(119890119894) be a monotonically

increasing function with its inverse function denoted as119866minus1119894

119867119894= 11989210158401015840

119894(119890119894) gt 0 119883

119894is a continuous differentiable and

invertible random variable which is independent of 119889119894 For

simplicity let 119883119894have the same probability density function

119891(119909) and cumulative distribution function 119865(119909) 119865(119909) isnonnegative strictly increasing and invertible and satisfies119865(0) = 0 and 119865(119909) = 1 minus 119865(119909) 119863 is total market demand119863 = 119863

1+ 1198632 which increases monotonically with the total

promotion levelThe product is perishable with a comparatively long

lead time and a short selling season This paper focuses onactivities from the beginning of the production season to theend of the selling season At the beginning of the productionseason retailer 119894 and the supplier sign an option contract withtwo parameters denoted as (119874

119894 119864119894) where 119874

119894is the option

price and 119864119894is the exercise price At the same time each

retailer purchases an option quantity denoted as 119902119894 at unit

price 119874119894 Then the supplier makes his production decision

based on the retailersrsquo option order quantities and begins toproduce In the selling season depending on actual demandretailers begin to exercise their option quantitiesTheunit saleprice of retailer 119894 is 119875

119894 and the unit production cost of the

supplier is119862 If the supplier fails to complete the option orderquantity exercised by the retailers then the retailers have theright to punish the supplier by shortage penalty cost denotedas119885119894 which is the cost to the supplier to obtain an additional

unit of product by expediting production or buying from analternative source

Without loss of generality it is assumed that the salvage ofthe supplier and retailers is zero and that there are no credit

losses for retailers in an out-of-stock situation To avoid trivialproblem and to ensure profit for all parties it is assumed that119862 lt 119874

119894+ 119864119894lt 119875119894 The notation 119909+ = max0 119909 will be used

3 Retailersrsquo Optimal Option Order Policy

At the beginning of the production season retailer 119894 pur-chases an option order quantity 119902

119894with an option contract

When retailer 119894 is risk-neutral the profit of retailer 119894 denotedas 120587119903119894(119902119894 119890119894 119863119894) is

120587119903

119894(119902119894 119890119894 119863119894) = 119875119894min (119902

119894 119863119894) minus 119874119894119902119894minus 119864119894min (119902

119894 119863119894)

minus 119892119894(119890119894)

(1)

In (1) the first term is the sales revenue the second term isthe option cost the third term is the exercise cost and thelast term is the promotional cost

Then the corresponding expected profit denoted as119864[120587119903

119894] is

119864 [120587119903

119894] = (119875

119894minus 119874119894minus 119864119894) 119902119894minus 119892119894(119890119894)

minus (119875119894minus 119864119894) int

119902119894minus119889119894

0

119865 (119909) 119889119909

(2)

Because retailer 119894 is risk-averse the degree of risk aversionshould be taken into account in determining the optionorder quantity In this paper retailer 119894 takes the CVaR as hisperformance measure because the CVaR risk measure is arelatively conservative decision-making criterion [35]

According to the definition given by Rockafellar andUryasev [33] the definition of the CVaR on retailer 119894rsquos optionorder quantity and promotion level is given by the following

Definition 1 120578119894-CVaR on retailer 119894rsquos option order quantity 119902

119894

and promotion level 119890119894 denoted as 119862

120578119894[120587119903

119894] is

119862120578119894[120587119903

119894] = 119864 120587

119903

119894(119902119894 119890119894 119863119894) | 120587119903

119894(119902119894 119890119894 119863119894)

⩽ 120573120578119894[120587119903

119894(119902119894 119890119894 119863119894)]

(3)

where 120573120578119894[120587119903

119894(119902119894 119890119894 119863119894)] = inf120573 | 119875[120587

119903

119894(119902119894 119890119894 119863119894) ⩽ 120573] ⩾ 120578

119894

is a quantile and 120578119894isin (0 1] reflects the degree of risk aversion

for retailer 119894 (the smaller 120578119894is the more risk-averse retailer 119894

is)

To facilitate the calculation an equivalent definition isgiven by [33 34]

119862120578119894[120587119903

119894] = max120572isinR

120572 minus1

120578119894

119864 [120572 minus 120587119903

119894(119902119894 119890119894 119863119894)]+ (4)

where 120572 is a real number Let

119880119894= 120572 minus

1

120578119894

119864 [120572 minus 120587119903

119894(119902119894 119890119894 119863119894)]+ (5)

Combining expressions (2) and (5) with some algebra yields

4 Mathematical Problems in Engineering

119880119894=

120572 120572 ⩽ 119882119894

120572 minus1

120578119894

int

(120572minus119882119894)119860119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 119882

119894lt 120572 ⩽ 119881

119894

120572 minus1

120578119894

int

119862119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 minus

1

120578119894

int

infin

119862119894

(120572 minus 119881119894) 119891 (119909) 119889119909 120572 gt 119881

119894

(6)

where 119860119894= 119875119894minus 119864119894 119861119894= 119875119894minus 119864119894minus 119874119894 119862119894= 119902119894minus 119889119894 119881119894=

(119875119894minus 119864119894minus 119874119894)119902119894minus 119892119894(119890119894) and119882

119894= (119875119894minus 119864119894)119889119894minus 119874119894119902119894minus 119892119894(119890119894)

Property 1 119862120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

The objective of retailer 119894 is to maximize his CVaRmeasure 119862

120578119894[120587119903

119894] The optimal solution of (3) denoted as

(119902lowast

119894 119890lowast

119894) can be obtained using (3) (6) and Property 1

Theorem 2 Given the degree of risk aversion 120578119894for retailer 119894

and the competitorrsquos promotion level 1198903minus119894

retailer 119894rsquos optimaloption ordering policy is given by

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(7)

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] =

119860119894

120578119894

int

119866120578119894

0

119865minus1(119905) 119889119905 + 119861119894119889119894 (119890

lowast

119894 1198903minus119894)

minus 119892119894(119890lowast

119894)

(8)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Theorem 2 shows that retailer 119894rsquos optimal option orderquantity contains two parts one is related to the promotionalcompetition and the other is determined by stochasticdemand This result is consistent with the form of demandfaced by retailers When retailer 119894 achieves the optimal levelof promotional activity 119890lowast

119894 retailer 119894rsquos option order quantity 119902lowast

119894

satisfies the following property

Property 2 Given the optimal promotion level 119890lowast119894for retailer

119894 then retailer 119894rsquos optimal option order quantity 119902lowast119894satisfies the

following properties 120597119902lowast119894120597119874119894lt 0 120597119902lowast

119894120597119864119894lt 0 120597119902lowast

119894120597120578119894gt 0

and 120597119902lowast1198941205971198903minus119894

lt 0

In Property 2 the first two items show that when theoption price and the exercise price rise retailer 119894 will reducehis cost by reducing the option order quantity to ensure hisprofitThe third item illustrates that the higher 120578

119894is the lower

retailer 119894rsquos degree of risk aversion will be making 119902lowast119894higher

This result is consistent with the intuition that risk-averseretailers would rather have a steady income than take a riskto obtain more benefit When 120578

119894= 1 retailer 119894 is risk-neutral

and retailer 119894rsquos optimal order policy (1199020lowast119894 1198900lowast

119894) is

1199020lowast

119894= 119889119894(1198900lowast

119894 1198903minus119894) + 119865minus1(1198660)

1198900lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(9)

where 1198660= 119861119894119860119894 Obviously 119902lowast

119894lt 1199020lowast

119894and 119890lowast119894lt 1198900lowast

119894 Wang

and Webster [12] derived a similar conclusion howevertheir result is based on a whole-price contract for a supplychain with a loss-averse retailer The last item in Property 2indicates that the higher the competitorrsquos promotion level isthe lower the retailer 119894rsquos optimal option order quantity willbe and corresponding profit will be reduced When retailer1 is in his best promotional environment the upper boundon 119902lowast1is obtained when 119890

2= 0 in (7) and the lower bound

on 119902lowast1is obtained when 119890

2rarr infin in (7) Therefore the curve

1199021= 119902lowast

1(119890lowast

1 1198902) is the reaction curve on (119890

1 1198902) for retailer 1

as illustrated in Figure 1If 119902lowast2(1198901 119890lowast

2) is retailer 2rsquos optimal option order quantity

similar conclusions will be obtained Figure 1 also illustratesthe reaction curve of retailer 2 that is the curve 119902

2=

119902lowast

2(1198901 119890lowast

2) As illustrated in Figure 1 there is an equilibrium

point between the two retailers At the equilibrium pointif each retailer knows the best level of promotional activityof the other then the two retailersrsquo competition satisfies thefollowing equilibrium equations

1198621205781[120587119903

1(119902lowast

1 elowast)] ⩾ 119862

1205781[120587119903

1(1199021 1198901 119890lowast

2)] forall119902

1 1198901⩾ 0

1198621205782[120587119903

2(119902lowast

2 elowast)] ⩾ 119862

1205782[120587119903

2(1199022 1198902 119890lowast

1)] forall119902

2 1198902⩾ 0

(10)

where elowast = (119890lowast1 119890lowast

2)

The next theorem shows the existence and uniqueness ofNash equilibrium between the two retailers

Theorem 3 Given that each of the two risk-averse retailersknows the best promotion level for the other then there existsa unique Nash equilibrium between the order quantities oftwo retailers engaging in promotion competition and theequilibrium satisfies the following conditions

119902lowast

119894= 119889lowast

119894+ 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(11)

where 119889lowast119894= 119889119894(119890lowast

119894 119890lowast

3minus119894)

Theorem 3 indicates that at the equilibrium point 119902lowast1=

119902lowast

2 it is possible to obtain the relationship of the two retailers

on sale price option price exercise price and degree of riskaversion for retailer 119894 that is119861

1= 1198612and 12057811205782= 11986011198602The

first item indicates that the two retailers have the same profitper unit of product (promotional cost is not considered) andthe second item shows that the two retailersrsquo degree of riskaversion is proportional to the difference between the saleprice and the exercise price (excluding the sale price option

Mathematical Problems in Engineering 5

e1

e2

qlowast

2(e

lowast

2 e1)

qlowast

1(e

lowast

1 e2)

Figure 1 Impact of the promotion level on equilibrium orderquantity

price and exercise price and degree of risk aversion of thetwo retailers is the same)

4 Supplierrsquos Optimal Production Decision

Before the production season the supplier will determinethe production quantity in accordance with the retailersrsquooption order quantity Considering that the two retailers willnot exercise all their options at the beginning of the sellingseason the supplier will reduce production by running therisk of being punished The supplierrsquos profit denoted by 120587119898is

120587119898=

2

sum

119894=1

120587119898

119894=

2

sum

119894=1

119874119894119902lowast

119894+ 119864119894min (119902lowast

119894 119863119894) minus 119862119902

119898

119894

minus 119885119894[min (119902lowast

119894 119863119894) minus 119902119898

119894]+

(12)

where 120587119898

119894is the profit from the supplierrsquos selling of the

production to retailer 119894 and 119902119898

119894is the supplierrsquos option

production for retailer 119894 On the right-hand side of the sumin (12) the first term is the option cost that retailer 119894 paysto the supplier the second term is the supplierrsquos revenuewhen retailers exercise their options the third term is theproduction cost and the last term is the shortage penalty costThe corresponding expected profit is

119864 [120587119898] =

2

sum

119894=1

119864 [120587119898

119894] =

2

sum

119894=1

[(119874119894+ 119864119894) 119902lowast

119894minus 119862119902119898

119894

minus 119864119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

minus 119885119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909]

(13)

Property 3 119864[120587119898] is a second-order differentiable function in119902119898

119894and satisfies the following properties

120597119864 [120587119898

119894]

120597119874119894

gt 0

120597119864 [120587119898

119894]

120597119864119894

gt 0

120597119864 [120587119898

119894]

120597119862lt 0

120597119864 [120587119898

119894]

120597119885119894

lt 0

(14)

Property 3 shows that the higher the option price and theexercise price are the higher 119864[120587119898

119894] will be and the higher

the production cost and the penalty cost are the lower 119864[120587119898119894]

will beThese results are consistent with our intuition and theactual situation

In the proof of Property 3 letting the first-order partialderivative of 119864[120587119898

119894] with respect to 119902119898

119894be equal to zero

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 (119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)) = 0 (15)

According to (15) the supplierrsquos production decision forretailer 119894 denoted as 119902119898Δ

119894 is given by

119902119898Δ

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866119898

119894) (16)

where 119866119898119894= (119885119894minus 119862)119885

119894

The first-order partial derivative of 119902119898Δ119894

with respect to119885119894

is 120597119902119898Δ119894120597119885119894= 119862119885

2

119894119891(119902119898Δ

119894minus 119889119894(119890lowast

119894 1198903minus119894)) gt 0 which implies

that for given 119862 119890lowast119894 and 119890

3minus119894 the supplierrsquos production is

increasing in 119885119894 In other words the supplier will choose to

produce more options to reduce the loss when the shortagepenalty cost increases But if the shortage penalty cost isgreater than a certain critical value the supplier will produceall the options or his marginal loss will be greater than themarginal profit Combining (7) and (15) yields the shortagepenalty threshold value

119885119894=

119860119894119862

119860119894minus 119861119894120578119894

(17)

The first-order partial derivative of 119885119894with respect to 120578

119894is

120597119885119894120597120578119894= 119860119894119861119894119862(119860119894minus 119861119894120578119894)2gt 0 which indicates that

the penalty threshold value 119885119894will be higher when retailer

119894 is not very risk-averse According to the above analysisit is clear that the penalty threshold value directly impactsthe supplierrsquos production decision The supplierrsquos optimalproduction decision for retailer 119894 is given by

119902119898lowast

119894=

119902119898Δ

119894 119885119894lt 119885119894

119902lowast

119894 119885

119894⩾ 119885119894

(18)

From (18) it is known thatwhen119885119894is less than119885

119894 the supplier

will accept the penalty and will produce only 119902119894Δ119898

units of

6 Mathematical Problems in Engineering

options for retailer 119894 otherwise he will produce all optionorder quantity At the same time the total optimal productionquantity 119902119898lowast of the supplier for two retailers is given by

119902119898lowast

=

119902119898Δ

1+ 119902119898Δ

2 1198851lt 1198851 1198852lt 1198852

119902119898Δ

1+ 119902lowast

2 119885

1lt 1198851 1198852⩾ 1198852

119902lowast

1+ 119902119898Δ

2 119885

1⩾ 1198851 1198852lt 1198852

119902lowast

1+ 119902lowast

2 119885

1⩾ 1198851 1198852⩾ 1198852

(19)

When 120578119894= 1 the penalty threshold value is119885

119894= 119860119894119862119874119894

and the corresponding optimal production decision 1199021198980lowast119894

ofthe supplier for retailer 119894 is

1199021198980lowast

119894=

1199021198980Δ

119894 119885119894lt 119885119894

1199020lowast

119894 119885

119894⩾ 119885119894

(20)

Obviously 119885119894lt 119885119894and 119902119898lowast119894

lt 1199021198980lowast

119894

5 Supply Chain Coordination

It is well known that the optimal decision of supply chainsystem-wide is the benchmark for supply chain coordinationTo derive the optimal decision of the channel the supplychain is taken as one entity and the profit of the supply chainsystem is formulated and denoted as 120587119904

120587119904=

2

sum

119894=1

[119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)] (21)

where 119902119904119894and 119890119904119894are respectively the order quantity and the

promotion level for retailer 119894 in the supply chain system Onthe right-hand side of the sum in (21) the first term is thesales revenue the second term is the production cost and thelast term is the promotion cost The corresponding expectedprofit is

119864 [120587119904] =

2

sum

119894=1

119864 [119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)]

=

2

sum

119894=1

[minus119875119894int

119902119904

119894minus119889119904

119894

0

119865 (119909) 119889119909 + (119875119894 minus 119862) 119902119904

119894minus 119892119894(119890119904

119894)]

(22)

Obviously the optimal decision of the supply chain canbe obtained by maximizing (22) Let the first-order partialderivative of 119864[120587119904] with respect to 119902119904

119894and 119890119904119894be equal to zero

120597119864 [120587119904]

120597119902119904

119894

= 119861119904

119894minus 119875119894119865 (119862119904

119894) = 0

120597119864 [120587119904]

120597119890119904

119894

= minus119866119894(119890119904

119894) + 119875119894119865 (119862119904

119894)119898119904

119894119894

+ 1198753minus119894119865 (119862119904

3minus119894)119898119904

3minus119894119894= 0

(23)

where 119862119904119894= 119902119904

119894minus 119889119904

119894 119894 = 1 2 In addition the leading

principleminors ofmatrix of119864[120587119904] are as followsminus119875119894119891(119862119904

119894) lt

0 119875119894119891(119862119904

119894)119867119894+ 1198753minus119894119891(119862119904

119894)119891(119862119904

3minus119894)(1198983minus119894119894

)2gt 0 The optimal

decision for the channel in (22) can be obtained as follows

119902119904lowast

119894= 119889119904lowast

119894+ 119865minus1(119866119904

119894)

119890119904lowast

119894= 119866minus1

119894(119898119904

119894119894119861119904

119894+ 119898119904

3minus119894119894119861119904

3minus119894)

(24)

where 119889119904lowast119894= 119889119894(119890119904lowast

119894 119890119904lowast

3minus119894) 119861119904119894= 119875119894minus 119862 119861119904

3minus119894= 1198753minus119894

minus 119862 119866119904119894=

119861119904

119894119875119894119898119904119894119894= 1198891015840

119894119890119904

119894

and1198981199043minus119894119894

= 1198891015840

3minus119894119890119904

119894

The optimal order quantity and the promotion level of the

supply chain system-wide can be used as the benchmark forsupply chain coordination to adjust the option parametersThe coordination conditions of supply chain will be given bythe following theorem

Theorem 4 The supply chain of a risk-neutral supplier andtwo risk-averse retailers engaging in promotion competitionwith an option contract and a CVaR criterion can be coordi-nated by the following conditions

1198611= 1198612

1205781

1205782

=1198601

1198602

1 minus119862

119875119894

lt 120578119894⩽ 1

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119885119894⩾ 119875119894

119862 lt 119874119894+ 119864119894lt 119875119894

(25)

In Theorem 4 the first two items show that if the supplychain can be coordinated then the two retailers first achievecompetitive equilibriumThe third item implies that retailersare not very risk-averse that is retailers will pursue profit bytaking some risk LetΔ119889

119894= 119889119894(119890119894+Δ119890119894 1198903minus119894)minus119889119894(119890119894 1198903minus119894)Δ119889119904119894=

119889119894(119890119904

119894+Δ119890119894 119890119904

3minus119894)minus119889119894(119890119904

119894 119890119904

3minus119894) andΔ119889119904

3minus119894= 1198893minus119894(119890119904

3minus119894 119890119904

119894+Δ119890119894)minus

1198893minus119894(119890119904

3minus119894 119890119904

119894) where Δ119890

119894is a small change on the promotion

level of retailer 119894 The fourth item shows that if the promotionlevel of retailer 119894 changes Δ119890

119894units then retailer 119894rsquos demand

will change Δ119889119904119894units and the other retailerrsquos demand will

change Δ1198891199043minus119894

units in the centralized case retailer 119894rsquos demandwill change (119861119904

119894Δ119889119904

119894+119861119904

3minus119894Δ119889119904

3minus119894)119861119894units in the decentralized

case This relationship is brought about by the two retailersrsquocompetition and promotional activities Furthermore thepenalty threshold value must be higher than the sale priceor the supplier will not produce all option order quantityFinally by adjusting the parameters of the option contractthe whole supply chain profit can reach the optimum andthe profits of supply chain members can achieve Paretooptimum

Mathematical Problems in Engineering 7

Table 1 Comparison of the coordination conditions of several supply chains

SC1

SC2

119878119862119894

1198781198623119894

119885 ⩾ 119875 119885 ⩾ 119875 119885119894⩾ 119875119894

119885119894⩾ 119875119894

119875 = 119864 +119875119874

119862119875 = 119864 +

119875119874120578

119862 minus 119875(1 minus 120578)119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119875119894= 119864119894+119875119894119874119894

119862

120578 gt 1 minus119862

119875120578119894gt 1 minus

119862

119875119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

S

S1

S2

R1

PromotionCompetitionR2

C1

C2

SC1

SC

SC2

Figure 2 The relationship of 119878119862 agents

Then we will discuss the subchain coordination condi-tions Figure 2 shows that the supply chain contains twosubchains and the coordination conditions of subchain willbe given by the following theorem

Theorem 5 The subchain (119878119862119894) can be coordinated by the

following conditions

120578119894gt 1 minus

119862

119875119894

119885119894⩾ 119875119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

gt 1

(26)

FromTheorem 5 we can see that the first three coordina-tion conditions of 119878119862

119894are similar to that of the entity supply

chain The last condition implies that Δ119889119894gt Δ119889

119904

119894 which

means that the level of the promotion activity of retailer 119894 indecentralized decision is higher than in centralized decision

Furthermore we can derive the coordination conditionsof supply chain with a neutral supplier and a neutral retailer(1198781198621) supply chain with a neutral supplier and a risk retailer

(1198781198622) and subchain of supply chain with a neutral supplier

and two neutral retailers engaging in promotion competition(1198781198623119894) In the coordination conditions of 119878119862

119894 when 120578

119894and 119890119894

are 1 and 0 respectively we get the coordination conditionsof 1198781198623119894and 119878119862

2 which are consistent with the conclusion of

[39] and when 120578119894= 1 and 119890

119894= 0 the coordination conditions

of 1198781198621are obtained Now we compare them in Table 1 From

Table 1 we find that the penalty cost is higher than theretail price for four supply chains or subchains coordination

conditions Obviously this condition is beneficial for theretailers to fully exercise the option and the supplier hasto produce all option order quantity to reduce loss Table 1shows that the sale price of 119878119862

1is higher than 119878119862

2 which

embodies the characteristic of the retailerrsquos risk aversionSimilar conclusions exist in 119878119862

119894and 119878119862

3119894

6 Numerical Analysis

In this section we carry out numerical experiments underthe model assumption to illustrate our findings We let 119875

1=

55 1198641= 335 119874

1= 58 119875

2= 547 119864

2= 326 119874

2=

67 1205781= 068 120578

2= 071 119862 = 275 and 119885

1= 60

and 1198852= 56 For simplicity we assume that the random

demand variable of each retailer is uniformly distributedon [0 300] and 119889(119890

119894 1198903minus119894) = 100 + 5119890

119894minus 043 sdot 5119890

3minus119894 The

retailersrsquo optimal option order policy and supplierrsquos optimalproduction decision in decentralized case and the optimalsupply chain decision system-wide are shown in Table 2 (notethat the data are rounded) In Table 2 retailersrsquo optimal orderquantity is the same as the supplierrsquos production decisionand the level of two retailersrsquo promotional activity remainsconsistent in decentralized case Furthermore the optimalprofit of supply chain system is the same as that of theretailers and the supplier in decentralized case which impliesthat the supply chain consisting of a risk-neutral supplierand two risk-averse retailers in competition and engagedin promotion is coordinated under the option contract andCVaR criterion

Thenwe analyze the impact of the level of the promotionalactivity on the retailersrsquo order quantities by fixing 119890lowast

1= 85

and 119890lowast2= 85 respectively and varying 119890

2and 1198901from 0 to

200 in steps of 5 corresponding 119890lowast1and 119890lowast2 Figure 3(a) shows

that retailer 1rsquos order quantity will increase when 1198901increases

however retailer 2rsquos order quantity will decrease similar toFigure 3(b) Figures 3(a) and 3(b) also illustrate the uniqueequilibrium point between two risk-averse retailers which isin agreement with the conclusion of Theorem 3

7 Conclusion

This paper investigates an option contract for coordinatinga supply chain with one risk-neutral supplier and two risk-averse retailers engaged in promotion competition Based onthe option contract the optimal option order quantity andthe promotion level of two retailers are obtained with CVaR

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

a CVaR criterion and the supplierrsquos optimal production deci-sion under an option contract The equilibrium competitionbetween two risk-averse retailers is also analyzed Section 4explores supply chain system and its subchain coordinationissues and the corresponding coordination conditions aregiven Section 5 illustrates the impact of the promotionlevel on the optimal order quantity of each retailer throughnumerical experiments Section 6 concludes the paper

2 Model Formulation and Assumptions

Let us consider a one-period two-echelon supply chaincoordination problemThe supply chain consists of one risk-neutral supplier and two risk-averse retailers engaged inpromotion competition In the traditional Cournot compe-tition the demand is considered as the determination [10]but in fact the demand is uncertain We assume that tworetailers order the product from the supplier with demanduncertainty The uncertain demand faced by retailer 119894 (119894 =1 2) is 119863

119894 which takes on an additive form and can be

expressed as 119863119894= 119889119894+ 119883119894 119889119894⩾ 0 is a demand relating

to the market scale and promotion level We assume that119889119894= 119889119894(119890119894 1198903minus119894) is a linear function increasing (decreasing)

monotonically with the retailerrsquos (the competitorrsquos) promo-tion level 119890

119894(1198903minus119894) which can be described by the amount of

promotional products Let 119892119894(119890119894) denote the promotional cost

of retailer 119894 which is a second-order differentiable functionsatisfying 119892

119894(0) = 0 Let 119866

119894= 1198921015840

119894(119890119894) be a monotonically

increasing function with its inverse function denoted as119866minus1119894

119867119894= 11989210158401015840

119894(119890119894) gt 0 119883

119894is a continuous differentiable and

invertible random variable which is independent of 119889119894 For

simplicity let 119883119894have the same probability density function

119891(119909) and cumulative distribution function 119865(119909) 119865(119909) isnonnegative strictly increasing and invertible and satisfies119865(0) = 0 and 119865(119909) = 1 minus 119865(119909) 119863 is total market demand119863 = 119863

1+ 1198632 which increases monotonically with the total

promotion levelThe product is perishable with a comparatively long

lead time and a short selling season This paper focuses onactivities from the beginning of the production season to theend of the selling season At the beginning of the productionseason retailer 119894 and the supplier sign an option contract withtwo parameters denoted as (119874

119894 119864119894) where 119874

119894is the option

price and 119864119894is the exercise price At the same time each

retailer purchases an option quantity denoted as 119902119894 at unit

price 119874119894 Then the supplier makes his production decision

based on the retailersrsquo option order quantities and begins toproduce In the selling season depending on actual demandretailers begin to exercise their option quantitiesTheunit saleprice of retailer 119894 is 119875

119894 and the unit production cost of the

supplier is119862 If the supplier fails to complete the option orderquantity exercised by the retailers then the retailers have theright to punish the supplier by shortage penalty cost denotedas119885119894 which is the cost to the supplier to obtain an additional

unit of product by expediting production or buying from analternative source

Without loss of generality it is assumed that the salvage ofthe supplier and retailers is zero and that there are no credit

losses for retailers in an out-of-stock situation To avoid trivialproblem and to ensure profit for all parties it is assumed that119862 lt 119874

119894+ 119864119894lt 119875119894 The notation 119909+ = max0 119909 will be used

3 Retailersrsquo Optimal Option Order Policy

At the beginning of the production season retailer 119894 pur-chases an option order quantity 119902

119894with an option contract

When retailer 119894 is risk-neutral the profit of retailer 119894 denotedas 120587119903119894(119902119894 119890119894 119863119894) is

120587119903

119894(119902119894 119890119894 119863119894) = 119875119894min (119902

119894 119863119894) minus 119874119894119902119894minus 119864119894min (119902

119894 119863119894)

minus 119892119894(119890119894)

(1)

In (1) the first term is the sales revenue the second term isthe option cost the third term is the exercise cost and thelast term is the promotional cost

Then the corresponding expected profit denoted as119864[120587119903

119894] is

119864 [120587119903

119894] = (119875

119894minus 119874119894minus 119864119894) 119902119894minus 119892119894(119890119894)

minus (119875119894minus 119864119894) int

119902119894minus119889119894

0

119865 (119909) 119889119909

(2)

Because retailer 119894 is risk-averse the degree of risk aversionshould be taken into account in determining the optionorder quantity In this paper retailer 119894 takes the CVaR as hisperformance measure because the CVaR risk measure is arelatively conservative decision-making criterion [35]

According to the definition given by Rockafellar andUryasev [33] the definition of the CVaR on retailer 119894rsquos optionorder quantity and promotion level is given by the following

Definition 1 120578119894-CVaR on retailer 119894rsquos option order quantity 119902

119894

and promotion level 119890119894 denoted as 119862

120578119894[120587119903

119894] is

119862120578119894[120587119903

119894] = 119864 120587

119903

119894(119902119894 119890119894 119863119894) | 120587119903

119894(119902119894 119890119894 119863119894)

⩽ 120573120578119894[120587119903

119894(119902119894 119890119894 119863119894)]

(3)

where 120573120578119894[120587119903

119894(119902119894 119890119894 119863119894)] = inf120573 | 119875[120587

119903

119894(119902119894 119890119894 119863119894) ⩽ 120573] ⩾ 120578

119894

is a quantile and 120578119894isin (0 1] reflects the degree of risk aversion

for retailer 119894 (the smaller 120578119894is the more risk-averse retailer 119894

is)

To facilitate the calculation an equivalent definition isgiven by [33 34]

119862120578119894[120587119903

119894] = max120572isinR

120572 minus1

120578119894

119864 [120572 minus 120587119903

119894(119902119894 119890119894 119863119894)]+ (4)

where 120572 is a real number Let

119880119894= 120572 minus

1

120578119894

119864 [120572 minus 120587119903

119894(119902119894 119890119894 119863119894)]+ (5)

Combining expressions (2) and (5) with some algebra yields

4 Mathematical Problems in Engineering

119880119894=

120572 120572 ⩽ 119882119894

120572 minus1

120578119894

int

(120572minus119882119894)119860119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 119882

119894lt 120572 ⩽ 119881

119894

120572 minus1

120578119894

int

119862119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 minus

1

120578119894

int

infin

119862119894

(120572 minus 119881119894) 119891 (119909) 119889119909 120572 gt 119881

119894

(6)

where 119860119894= 119875119894minus 119864119894 119861119894= 119875119894minus 119864119894minus 119874119894 119862119894= 119902119894minus 119889119894 119881119894=

(119875119894minus 119864119894minus 119874119894)119902119894minus 119892119894(119890119894) and119882

119894= (119875119894minus 119864119894)119889119894minus 119874119894119902119894minus 119892119894(119890119894)

Property 1 119862120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

The objective of retailer 119894 is to maximize his CVaRmeasure 119862

120578119894[120587119903

119894] The optimal solution of (3) denoted as

(119902lowast

119894 119890lowast

119894) can be obtained using (3) (6) and Property 1

Theorem 2 Given the degree of risk aversion 120578119894for retailer 119894

and the competitorrsquos promotion level 1198903minus119894

retailer 119894rsquos optimaloption ordering policy is given by

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(7)

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] =

119860119894

120578119894

int

119866120578119894

0

119865minus1(119905) 119889119905 + 119861119894119889119894 (119890

lowast

119894 1198903minus119894)

minus 119892119894(119890lowast

119894)

(8)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Theorem 2 shows that retailer 119894rsquos optimal option orderquantity contains two parts one is related to the promotionalcompetition and the other is determined by stochasticdemand This result is consistent with the form of demandfaced by retailers When retailer 119894 achieves the optimal levelof promotional activity 119890lowast

119894 retailer 119894rsquos option order quantity 119902lowast

119894

satisfies the following property

Property 2 Given the optimal promotion level 119890lowast119894for retailer

119894 then retailer 119894rsquos optimal option order quantity 119902lowast119894satisfies the

following properties 120597119902lowast119894120597119874119894lt 0 120597119902lowast

119894120597119864119894lt 0 120597119902lowast

119894120597120578119894gt 0

and 120597119902lowast1198941205971198903minus119894

lt 0

In Property 2 the first two items show that when theoption price and the exercise price rise retailer 119894 will reducehis cost by reducing the option order quantity to ensure hisprofitThe third item illustrates that the higher 120578

119894is the lower

retailer 119894rsquos degree of risk aversion will be making 119902lowast119894higher

This result is consistent with the intuition that risk-averseretailers would rather have a steady income than take a riskto obtain more benefit When 120578

119894= 1 retailer 119894 is risk-neutral

and retailer 119894rsquos optimal order policy (1199020lowast119894 1198900lowast

119894) is

1199020lowast

119894= 119889119894(1198900lowast

119894 1198903minus119894) + 119865minus1(1198660)

1198900lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(9)

where 1198660= 119861119894119860119894 Obviously 119902lowast

119894lt 1199020lowast

119894and 119890lowast119894lt 1198900lowast

119894 Wang

and Webster [12] derived a similar conclusion howevertheir result is based on a whole-price contract for a supplychain with a loss-averse retailer The last item in Property 2indicates that the higher the competitorrsquos promotion level isthe lower the retailer 119894rsquos optimal option order quantity willbe and corresponding profit will be reduced When retailer1 is in his best promotional environment the upper boundon 119902lowast1is obtained when 119890

2= 0 in (7) and the lower bound

on 119902lowast1is obtained when 119890

2rarr infin in (7) Therefore the curve

1199021= 119902lowast

1(119890lowast

1 1198902) is the reaction curve on (119890

1 1198902) for retailer 1

as illustrated in Figure 1If 119902lowast2(1198901 119890lowast

2) is retailer 2rsquos optimal option order quantity

similar conclusions will be obtained Figure 1 also illustratesthe reaction curve of retailer 2 that is the curve 119902

2=

119902lowast

2(1198901 119890lowast

2) As illustrated in Figure 1 there is an equilibrium

point between the two retailers At the equilibrium pointif each retailer knows the best level of promotional activityof the other then the two retailersrsquo competition satisfies thefollowing equilibrium equations

1198621205781[120587119903

1(119902lowast

1 elowast)] ⩾ 119862

1205781[120587119903

1(1199021 1198901 119890lowast

2)] forall119902

1 1198901⩾ 0

1198621205782[120587119903

2(119902lowast

2 elowast)] ⩾ 119862

1205782[120587119903

2(1199022 1198902 119890lowast

1)] forall119902

2 1198902⩾ 0

(10)

where elowast = (119890lowast1 119890lowast

2)

The next theorem shows the existence and uniqueness ofNash equilibrium between the two retailers

Theorem 3 Given that each of the two risk-averse retailersknows the best promotion level for the other then there existsa unique Nash equilibrium between the order quantities oftwo retailers engaging in promotion competition and theequilibrium satisfies the following conditions

119902lowast

119894= 119889lowast

119894+ 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(11)

where 119889lowast119894= 119889119894(119890lowast

119894 119890lowast

3minus119894)

Theorem 3 indicates that at the equilibrium point 119902lowast1=

119902lowast

2 it is possible to obtain the relationship of the two retailers

on sale price option price exercise price and degree of riskaversion for retailer 119894 that is119861

1= 1198612and 12057811205782= 11986011198602The

first item indicates that the two retailers have the same profitper unit of product (promotional cost is not considered) andthe second item shows that the two retailersrsquo degree of riskaversion is proportional to the difference between the saleprice and the exercise price (excluding the sale price option

Mathematical Problems in Engineering 5

e1

e2

qlowast

2(e

lowast

2 e1)

qlowast

1(e

lowast

1 e2)

Figure 1 Impact of the promotion level on equilibrium orderquantity

price and exercise price and degree of risk aversion of thetwo retailers is the same)

4 Supplierrsquos Optimal Production Decision

Before the production season the supplier will determinethe production quantity in accordance with the retailersrsquooption order quantity Considering that the two retailers willnot exercise all their options at the beginning of the sellingseason the supplier will reduce production by running therisk of being punished The supplierrsquos profit denoted by 120587119898is

120587119898=

2

sum

119894=1

120587119898

119894=

2

sum

119894=1

119874119894119902lowast

119894+ 119864119894min (119902lowast

119894 119863119894) minus 119862119902

119898

119894

minus 119885119894[min (119902lowast

119894 119863119894) minus 119902119898

119894]+

(12)

where 120587119898

119894is the profit from the supplierrsquos selling of the

production to retailer 119894 and 119902119898

119894is the supplierrsquos option

production for retailer 119894 On the right-hand side of the sumin (12) the first term is the option cost that retailer 119894 paysto the supplier the second term is the supplierrsquos revenuewhen retailers exercise their options the third term is theproduction cost and the last term is the shortage penalty costThe corresponding expected profit is

119864 [120587119898] =

2

sum

119894=1

119864 [120587119898

119894] =

2

sum

119894=1

[(119874119894+ 119864119894) 119902lowast

119894minus 119862119902119898

119894

minus 119864119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

minus 119885119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909]

(13)

Property 3 119864[120587119898] is a second-order differentiable function in119902119898

119894and satisfies the following properties

120597119864 [120587119898

119894]

120597119874119894

gt 0

120597119864 [120587119898

119894]

120597119864119894

gt 0

120597119864 [120587119898

119894]

120597119862lt 0

120597119864 [120587119898

119894]

120597119885119894

lt 0

(14)

Property 3 shows that the higher the option price and theexercise price are the higher 119864[120587119898

119894] will be and the higher

the production cost and the penalty cost are the lower 119864[120587119898119894]

will beThese results are consistent with our intuition and theactual situation

In the proof of Property 3 letting the first-order partialderivative of 119864[120587119898

119894] with respect to 119902119898

119894be equal to zero

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 (119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)) = 0 (15)

According to (15) the supplierrsquos production decision forretailer 119894 denoted as 119902119898Δ

119894 is given by

119902119898Δ

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866119898

119894) (16)

where 119866119898119894= (119885119894minus 119862)119885

119894

The first-order partial derivative of 119902119898Δ119894

with respect to119885119894

is 120597119902119898Δ119894120597119885119894= 119862119885

2

119894119891(119902119898Δ

119894minus 119889119894(119890lowast

119894 1198903minus119894)) gt 0 which implies

that for given 119862 119890lowast119894 and 119890

3minus119894 the supplierrsquos production is

increasing in 119885119894 In other words the supplier will choose to

produce more options to reduce the loss when the shortagepenalty cost increases But if the shortage penalty cost isgreater than a certain critical value the supplier will produceall the options or his marginal loss will be greater than themarginal profit Combining (7) and (15) yields the shortagepenalty threshold value

119885119894=

119860119894119862

119860119894minus 119861119894120578119894

(17)

The first-order partial derivative of 119885119894with respect to 120578

119894is

120597119885119894120597120578119894= 119860119894119861119894119862(119860119894minus 119861119894120578119894)2gt 0 which indicates that

the penalty threshold value 119885119894will be higher when retailer

119894 is not very risk-averse According to the above analysisit is clear that the penalty threshold value directly impactsthe supplierrsquos production decision The supplierrsquos optimalproduction decision for retailer 119894 is given by

119902119898lowast

119894=

119902119898Δ

119894 119885119894lt 119885119894

119902lowast

119894 119885

119894⩾ 119885119894

(18)

From (18) it is known thatwhen119885119894is less than119885

119894 the supplier

will accept the penalty and will produce only 119902119894Δ119898

units of

6 Mathematical Problems in Engineering

options for retailer 119894 otherwise he will produce all optionorder quantity At the same time the total optimal productionquantity 119902119898lowast of the supplier for two retailers is given by

119902119898lowast

=

119902119898Δ

1+ 119902119898Δ

2 1198851lt 1198851 1198852lt 1198852

119902119898Δ

1+ 119902lowast

2 119885

1lt 1198851 1198852⩾ 1198852

119902lowast

1+ 119902119898Δ

2 119885

1⩾ 1198851 1198852lt 1198852

119902lowast

1+ 119902lowast

2 119885

1⩾ 1198851 1198852⩾ 1198852

(19)

When 120578119894= 1 the penalty threshold value is119885

119894= 119860119894119862119874119894

and the corresponding optimal production decision 1199021198980lowast119894

ofthe supplier for retailer 119894 is

1199021198980lowast

119894=

1199021198980Δ

119894 119885119894lt 119885119894

1199020lowast

119894 119885

119894⩾ 119885119894

(20)

Obviously 119885119894lt 119885119894and 119902119898lowast119894

lt 1199021198980lowast

119894

5 Supply Chain Coordination

It is well known that the optimal decision of supply chainsystem-wide is the benchmark for supply chain coordinationTo derive the optimal decision of the channel the supplychain is taken as one entity and the profit of the supply chainsystem is formulated and denoted as 120587119904

120587119904=

2

sum

119894=1

[119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)] (21)

where 119902119904119894and 119890119904119894are respectively the order quantity and the

promotion level for retailer 119894 in the supply chain system Onthe right-hand side of the sum in (21) the first term is thesales revenue the second term is the production cost and thelast term is the promotion cost The corresponding expectedprofit is

119864 [120587119904] =

2

sum

119894=1

119864 [119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)]

=

2

sum

119894=1

[minus119875119894int

119902119904

119894minus119889119904

119894

0

119865 (119909) 119889119909 + (119875119894 minus 119862) 119902119904

119894minus 119892119894(119890119904

119894)]

(22)

Obviously the optimal decision of the supply chain canbe obtained by maximizing (22) Let the first-order partialderivative of 119864[120587119904] with respect to 119902119904

119894and 119890119904119894be equal to zero

120597119864 [120587119904]

120597119902119904

119894

= 119861119904

119894minus 119875119894119865 (119862119904

119894) = 0

120597119864 [120587119904]

120597119890119904

119894

= minus119866119894(119890119904

119894) + 119875119894119865 (119862119904

119894)119898119904

119894119894

+ 1198753minus119894119865 (119862119904

3minus119894)119898119904

3minus119894119894= 0

(23)

where 119862119904119894= 119902119904

119894minus 119889119904

119894 119894 = 1 2 In addition the leading

principleminors ofmatrix of119864[120587119904] are as followsminus119875119894119891(119862119904

119894) lt

0 119875119894119891(119862119904

119894)119867119894+ 1198753minus119894119891(119862119904

119894)119891(119862119904

3minus119894)(1198983minus119894119894

)2gt 0 The optimal

decision for the channel in (22) can be obtained as follows

119902119904lowast

119894= 119889119904lowast

119894+ 119865minus1(119866119904

119894)

119890119904lowast

119894= 119866minus1

119894(119898119904

119894119894119861119904

119894+ 119898119904

3minus119894119894119861119904

3minus119894)

(24)

where 119889119904lowast119894= 119889119894(119890119904lowast

119894 119890119904lowast

3minus119894) 119861119904119894= 119875119894minus 119862 119861119904

3minus119894= 1198753minus119894

minus 119862 119866119904119894=

119861119904

119894119875119894119898119904119894119894= 1198891015840

119894119890119904

119894

and1198981199043minus119894119894

= 1198891015840

3minus119894119890119904

119894

The optimal order quantity and the promotion level of the

supply chain system-wide can be used as the benchmark forsupply chain coordination to adjust the option parametersThe coordination conditions of supply chain will be given bythe following theorem

Theorem 4 The supply chain of a risk-neutral supplier andtwo risk-averse retailers engaging in promotion competitionwith an option contract and a CVaR criterion can be coordi-nated by the following conditions

1198611= 1198612

1205781

1205782

=1198601

1198602

1 minus119862

119875119894

lt 120578119894⩽ 1

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119885119894⩾ 119875119894

119862 lt 119874119894+ 119864119894lt 119875119894

(25)

In Theorem 4 the first two items show that if the supplychain can be coordinated then the two retailers first achievecompetitive equilibriumThe third item implies that retailersare not very risk-averse that is retailers will pursue profit bytaking some risk LetΔ119889

119894= 119889119894(119890119894+Δ119890119894 1198903minus119894)minus119889119894(119890119894 1198903minus119894)Δ119889119904119894=

119889119894(119890119904

119894+Δ119890119894 119890119904

3minus119894)minus119889119894(119890119904

119894 119890119904

3minus119894) andΔ119889119904

3minus119894= 1198893minus119894(119890119904

3minus119894 119890119904

119894+Δ119890119894)minus

1198893minus119894(119890119904

3minus119894 119890119904

119894) where Δ119890

119894is a small change on the promotion

level of retailer 119894 The fourth item shows that if the promotionlevel of retailer 119894 changes Δ119890

119894units then retailer 119894rsquos demand

will change Δ119889119904119894units and the other retailerrsquos demand will

change Δ1198891199043minus119894

units in the centralized case retailer 119894rsquos demandwill change (119861119904

119894Δ119889119904

119894+119861119904

3minus119894Δ119889119904

3minus119894)119861119894units in the decentralized

case This relationship is brought about by the two retailersrsquocompetition and promotional activities Furthermore thepenalty threshold value must be higher than the sale priceor the supplier will not produce all option order quantityFinally by adjusting the parameters of the option contractthe whole supply chain profit can reach the optimum andthe profits of supply chain members can achieve Paretooptimum

Mathematical Problems in Engineering 7

Table 1 Comparison of the coordination conditions of several supply chains

SC1

SC2

119878119862119894

1198781198623119894

119885 ⩾ 119875 119885 ⩾ 119875 119885119894⩾ 119875119894

119885119894⩾ 119875119894

119875 = 119864 +119875119874

119862119875 = 119864 +

119875119874120578

119862 minus 119875(1 minus 120578)119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119875119894= 119864119894+119875119894119874119894

119862

120578 gt 1 minus119862

119875120578119894gt 1 minus

119862

119875119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

S

S1

S2

R1

PromotionCompetitionR2

C1

C2

SC1

SC

SC2

Figure 2 The relationship of 119878119862 agents

Then we will discuss the subchain coordination condi-tions Figure 2 shows that the supply chain contains twosubchains and the coordination conditions of subchain willbe given by the following theorem

Theorem 5 The subchain (119878119862119894) can be coordinated by the

following conditions

120578119894gt 1 minus

119862

119875119894

119885119894⩾ 119875119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

gt 1

(26)

FromTheorem 5 we can see that the first three coordina-tion conditions of 119878119862

119894are similar to that of the entity supply

chain The last condition implies that Δ119889119894gt Δ119889

119904

119894 which

means that the level of the promotion activity of retailer 119894 indecentralized decision is higher than in centralized decision

Furthermore we can derive the coordination conditionsof supply chain with a neutral supplier and a neutral retailer(1198781198621) supply chain with a neutral supplier and a risk retailer

(1198781198622) and subchain of supply chain with a neutral supplier

and two neutral retailers engaging in promotion competition(1198781198623119894) In the coordination conditions of 119878119862

119894 when 120578

119894and 119890119894

are 1 and 0 respectively we get the coordination conditionsof 1198781198623119894and 119878119862

2 which are consistent with the conclusion of

[39] and when 120578119894= 1 and 119890

119894= 0 the coordination conditions

of 1198781198621are obtained Now we compare them in Table 1 From

Table 1 we find that the penalty cost is higher than theretail price for four supply chains or subchains coordination

conditions Obviously this condition is beneficial for theretailers to fully exercise the option and the supplier hasto produce all option order quantity to reduce loss Table 1shows that the sale price of 119878119862

1is higher than 119878119862

2 which

embodies the characteristic of the retailerrsquos risk aversionSimilar conclusions exist in 119878119862

119894and 119878119862

3119894

6 Numerical Analysis

In this section we carry out numerical experiments underthe model assumption to illustrate our findings We let 119875

1=

55 1198641= 335 119874

1= 58 119875

2= 547 119864

2= 326 119874

2=

67 1205781= 068 120578

2= 071 119862 = 275 and 119885

1= 60

and 1198852= 56 For simplicity we assume that the random

demand variable of each retailer is uniformly distributedon [0 300] and 119889(119890

119894 1198903minus119894) = 100 + 5119890

119894minus 043 sdot 5119890

3minus119894 The

retailersrsquo optimal option order policy and supplierrsquos optimalproduction decision in decentralized case and the optimalsupply chain decision system-wide are shown in Table 2 (notethat the data are rounded) In Table 2 retailersrsquo optimal orderquantity is the same as the supplierrsquos production decisionand the level of two retailersrsquo promotional activity remainsconsistent in decentralized case Furthermore the optimalprofit of supply chain system is the same as that of theretailers and the supplier in decentralized case which impliesthat the supply chain consisting of a risk-neutral supplierand two risk-averse retailers in competition and engagedin promotion is coordinated under the option contract andCVaR criterion

Thenwe analyze the impact of the level of the promotionalactivity on the retailersrsquo order quantities by fixing 119890lowast

1= 85

and 119890lowast2= 85 respectively and varying 119890

2and 1198901from 0 to

200 in steps of 5 corresponding 119890lowast1and 119890lowast2 Figure 3(a) shows

that retailer 1rsquos order quantity will increase when 1198901increases

however retailer 2rsquos order quantity will decrease similar toFigure 3(b) Figures 3(a) and 3(b) also illustrate the uniqueequilibrium point between two risk-averse retailers which isin agreement with the conclusion of Theorem 3

7 Conclusion

This paper investigates an option contract for coordinatinga supply chain with one risk-neutral supplier and two risk-averse retailers engaged in promotion competition Based onthe option contract the optimal option order quantity andthe promotion level of two retailers are obtained with CVaR

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

119880119894=

120572 120572 ⩽ 119882119894

120572 minus1

120578119894

int

(120572minus119882119894)119860119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 119882

119894lt 120572 ⩽ 119881

119894

120572 minus1

120578119894

int

119862119894

0

(120572 minus119882119894minus 119860119894119909)119891 (119909) 119889119909 minus

1

120578119894

int

infin

119862119894

(120572 minus 119881119894) 119891 (119909) 119889119909 120572 gt 119881

119894

(6)

where 119860119894= 119875119894minus 119864119894 119861119894= 119875119894minus 119864119894minus 119874119894 119862119894= 119902119894minus 119889119894 119881119894=

(119875119894minus 119864119894minus 119874119894)119902119894minus 119892119894(119890119894) and119882

119894= (119875119894minus 119864119894)119889119894minus 119874119894119902119894minus 119892119894(119890119894)

Property 1 119862120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

The objective of retailer 119894 is to maximize his CVaRmeasure 119862

120578119894[120587119903

119894] The optimal solution of (3) denoted as

(119902lowast

119894 119890lowast

119894) can be obtained using (3) (6) and Property 1

Theorem 2 Given the degree of risk aversion 120578119894for retailer 119894

and the competitorrsquos promotion level 1198903minus119894

retailer 119894rsquos optimaloption ordering policy is given by

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(7)

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] =

119860119894

120578119894

int

119866120578119894

0

119865minus1(119905) 119889119905 + 119861119894119889119894 (119890

lowast

119894 1198903minus119894)

minus 119892119894(119890lowast

119894)

(8)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Theorem 2 shows that retailer 119894rsquos optimal option orderquantity contains two parts one is related to the promotionalcompetition and the other is determined by stochasticdemand This result is consistent with the form of demandfaced by retailers When retailer 119894 achieves the optimal levelof promotional activity 119890lowast

119894 retailer 119894rsquos option order quantity 119902lowast

119894

satisfies the following property

Property 2 Given the optimal promotion level 119890lowast119894for retailer

119894 then retailer 119894rsquos optimal option order quantity 119902lowast119894satisfies the

following properties 120597119902lowast119894120597119874119894lt 0 120597119902lowast

119894120597119864119894lt 0 120597119902lowast

119894120597120578119894gt 0

and 120597119902lowast1198941205971198903minus119894

lt 0

In Property 2 the first two items show that when theoption price and the exercise price rise retailer 119894 will reducehis cost by reducing the option order quantity to ensure hisprofitThe third item illustrates that the higher 120578

119894is the lower

retailer 119894rsquos degree of risk aversion will be making 119902lowast119894higher

This result is consistent with the intuition that risk-averseretailers would rather have a steady income than take a riskto obtain more benefit When 120578

119894= 1 retailer 119894 is risk-neutral

and retailer 119894rsquos optimal order policy (1199020lowast119894 1198900lowast

119894) is

1199020lowast

119894= 119889119894(1198900lowast

119894 1198903minus119894) + 119865minus1(1198660)

1198900lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(9)

where 1198660= 119861119894119860119894 Obviously 119902lowast

119894lt 1199020lowast

119894and 119890lowast119894lt 1198900lowast

119894 Wang

and Webster [12] derived a similar conclusion howevertheir result is based on a whole-price contract for a supplychain with a loss-averse retailer The last item in Property 2indicates that the higher the competitorrsquos promotion level isthe lower the retailer 119894rsquos optimal option order quantity willbe and corresponding profit will be reduced When retailer1 is in his best promotional environment the upper boundon 119902lowast1is obtained when 119890

2= 0 in (7) and the lower bound

on 119902lowast1is obtained when 119890

2rarr infin in (7) Therefore the curve

1199021= 119902lowast

1(119890lowast

1 1198902) is the reaction curve on (119890

1 1198902) for retailer 1

as illustrated in Figure 1If 119902lowast2(1198901 119890lowast

2) is retailer 2rsquos optimal option order quantity

similar conclusions will be obtained Figure 1 also illustratesthe reaction curve of retailer 2 that is the curve 119902

2=

119902lowast

2(1198901 119890lowast

2) As illustrated in Figure 1 there is an equilibrium

point between the two retailers At the equilibrium pointif each retailer knows the best level of promotional activityof the other then the two retailersrsquo competition satisfies thefollowing equilibrium equations

1198621205781[120587119903

1(119902lowast

1 elowast)] ⩾ 119862

1205781[120587119903

1(1199021 1198901 119890lowast

2)] forall119902

1 1198901⩾ 0

1198621205782[120587119903

2(119902lowast

2 elowast)] ⩾ 119862

1205782[120587119903

2(1199022 1198902 119890lowast

1)] forall119902

2 1198902⩾ 0

(10)

where elowast = (119890lowast1 119890lowast

2)

The next theorem shows the existence and uniqueness ofNash equilibrium between the two retailers

Theorem 3 Given that each of the two risk-averse retailersknows the best promotion level for the other then there existsa unique Nash equilibrium between the order quantities oftwo retailers engaging in promotion competition and theequilibrium satisfies the following conditions

119902lowast

119894= 119889lowast

119894+ 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119861119894119898119894119894)

(11)

where 119889lowast119894= 119889119894(119890lowast

119894 119890lowast

3minus119894)

Theorem 3 indicates that at the equilibrium point 119902lowast1=

119902lowast

2 it is possible to obtain the relationship of the two retailers

on sale price option price exercise price and degree of riskaversion for retailer 119894 that is119861

1= 1198612and 12057811205782= 11986011198602The

first item indicates that the two retailers have the same profitper unit of product (promotional cost is not considered) andthe second item shows that the two retailersrsquo degree of riskaversion is proportional to the difference between the saleprice and the exercise price (excluding the sale price option

Mathematical Problems in Engineering 5

e1

e2

qlowast

2(e

lowast

2 e1)

qlowast

1(e

lowast

1 e2)

Figure 1 Impact of the promotion level on equilibrium orderquantity

price and exercise price and degree of risk aversion of thetwo retailers is the same)

4 Supplierrsquos Optimal Production Decision

Before the production season the supplier will determinethe production quantity in accordance with the retailersrsquooption order quantity Considering that the two retailers willnot exercise all their options at the beginning of the sellingseason the supplier will reduce production by running therisk of being punished The supplierrsquos profit denoted by 120587119898is

120587119898=

2

sum

119894=1

120587119898

119894=

2

sum

119894=1

119874119894119902lowast

119894+ 119864119894min (119902lowast

119894 119863119894) minus 119862119902

119898

119894

minus 119885119894[min (119902lowast

119894 119863119894) minus 119902119898

119894]+

(12)

where 120587119898

119894is the profit from the supplierrsquos selling of the

production to retailer 119894 and 119902119898

119894is the supplierrsquos option

production for retailer 119894 On the right-hand side of the sumin (12) the first term is the option cost that retailer 119894 paysto the supplier the second term is the supplierrsquos revenuewhen retailers exercise their options the third term is theproduction cost and the last term is the shortage penalty costThe corresponding expected profit is

119864 [120587119898] =

2

sum

119894=1

119864 [120587119898

119894] =

2

sum

119894=1

[(119874119894+ 119864119894) 119902lowast

119894minus 119862119902119898

119894

minus 119864119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

minus 119885119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909]

(13)

Property 3 119864[120587119898] is a second-order differentiable function in119902119898

119894and satisfies the following properties

120597119864 [120587119898

119894]

120597119874119894

gt 0

120597119864 [120587119898

119894]

120597119864119894

gt 0

120597119864 [120587119898

119894]

120597119862lt 0

120597119864 [120587119898

119894]

120597119885119894

lt 0

(14)

Property 3 shows that the higher the option price and theexercise price are the higher 119864[120587119898

119894] will be and the higher

the production cost and the penalty cost are the lower 119864[120587119898119894]

will beThese results are consistent with our intuition and theactual situation

In the proof of Property 3 letting the first-order partialderivative of 119864[120587119898

119894] with respect to 119902119898

119894be equal to zero

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 (119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)) = 0 (15)

According to (15) the supplierrsquos production decision forretailer 119894 denoted as 119902119898Δ

119894 is given by

119902119898Δ

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866119898

119894) (16)

where 119866119898119894= (119885119894minus 119862)119885

119894

The first-order partial derivative of 119902119898Δ119894

with respect to119885119894

is 120597119902119898Δ119894120597119885119894= 119862119885

2

119894119891(119902119898Δ

119894minus 119889119894(119890lowast

119894 1198903minus119894)) gt 0 which implies

that for given 119862 119890lowast119894 and 119890

3minus119894 the supplierrsquos production is

increasing in 119885119894 In other words the supplier will choose to

produce more options to reduce the loss when the shortagepenalty cost increases But if the shortage penalty cost isgreater than a certain critical value the supplier will produceall the options or his marginal loss will be greater than themarginal profit Combining (7) and (15) yields the shortagepenalty threshold value

119885119894=

119860119894119862

119860119894minus 119861119894120578119894

(17)

The first-order partial derivative of 119885119894with respect to 120578

119894is

120597119885119894120597120578119894= 119860119894119861119894119862(119860119894minus 119861119894120578119894)2gt 0 which indicates that

the penalty threshold value 119885119894will be higher when retailer

119894 is not very risk-averse According to the above analysisit is clear that the penalty threshold value directly impactsthe supplierrsquos production decision The supplierrsquos optimalproduction decision for retailer 119894 is given by

119902119898lowast

119894=

119902119898Δ

119894 119885119894lt 119885119894

119902lowast

119894 119885

119894⩾ 119885119894

(18)

From (18) it is known thatwhen119885119894is less than119885

119894 the supplier

will accept the penalty and will produce only 119902119894Δ119898

units of

6 Mathematical Problems in Engineering

options for retailer 119894 otherwise he will produce all optionorder quantity At the same time the total optimal productionquantity 119902119898lowast of the supplier for two retailers is given by

119902119898lowast

=

119902119898Δ

1+ 119902119898Δ

2 1198851lt 1198851 1198852lt 1198852

119902119898Δ

1+ 119902lowast

2 119885

1lt 1198851 1198852⩾ 1198852

119902lowast

1+ 119902119898Δ

2 119885

1⩾ 1198851 1198852lt 1198852

119902lowast

1+ 119902lowast

2 119885

1⩾ 1198851 1198852⩾ 1198852

(19)

When 120578119894= 1 the penalty threshold value is119885

119894= 119860119894119862119874119894

and the corresponding optimal production decision 1199021198980lowast119894

ofthe supplier for retailer 119894 is

1199021198980lowast

119894=

1199021198980Δ

119894 119885119894lt 119885119894

1199020lowast

119894 119885

119894⩾ 119885119894

(20)

Obviously 119885119894lt 119885119894and 119902119898lowast119894

lt 1199021198980lowast

119894

5 Supply Chain Coordination

It is well known that the optimal decision of supply chainsystem-wide is the benchmark for supply chain coordinationTo derive the optimal decision of the channel the supplychain is taken as one entity and the profit of the supply chainsystem is formulated and denoted as 120587119904

120587119904=

2

sum

119894=1

[119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)] (21)

where 119902119904119894and 119890119904119894are respectively the order quantity and the

promotion level for retailer 119894 in the supply chain system Onthe right-hand side of the sum in (21) the first term is thesales revenue the second term is the production cost and thelast term is the promotion cost The corresponding expectedprofit is

119864 [120587119904] =

2

sum

119894=1

119864 [119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)]

=

2

sum

119894=1

[minus119875119894int

119902119904

119894minus119889119904

119894

0

119865 (119909) 119889119909 + (119875119894 minus 119862) 119902119904

119894minus 119892119894(119890119904

119894)]

(22)

Obviously the optimal decision of the supply chain canbe obtained by maximizing (22) Let the first-order partialderivative of 119864[120587119904] with respect to 119902119904

119894and 119890119904119894be equal to zero

120597119864 [120587119904]

120597119902119904

119894

= 119861119904

119894minus 119875119894119865 (119862119904

119894) = 0

120597119864 [120587119904]

120597119890119904

119894

= minus119866119894(119890119904

119894) + 119875119894119865 (119862119904

119894)119898119904

119894119894

+ 1198753minus119894119865 (119862119904

3minus119894)119898119904

3minus119894119894= 0

(23)

where 119862119904119894= 119902119904

119894minus 119889119904

119894 119894 = 1 2 In addition the leading

principleminors ofmatrix of119864[120587119904] are as followsminus119875119894119891(119862119904

119894) lt

0 119875119894119891(119862119904

119894)119867119894+ 1198753minus119894119891(119862119904

119894)119891(119862119904

3minus119894)(1198983minus119894119894

)2gt 0 The optimal

decision for the channel in (22) can be obtained as follows

119902119904lowast

119894= 119889119904lowast

119894+ 119865minus1(119866119904

119894)

119890119904lowast

119894= 119866minus1

119894(119898119904

119894119894119861119904

119894+ 119898119904

3minus119894119894119861119904

3minus119894)

(24)

where 119889119904lowast119894= 119889119894(119890119904lowast

119894 119890119904lowast

3minus119894) 119861119904119894= 119875119894minus 119862 119861119904

3minus119894= 1198753minus119894

minus 119862 119866119904119894=

119861119904

119894119875119894119898119904119894119894= 1198891015840

119894119890119904

119894

and1198981199043minus119894119894

= 1198891015840

3minus119894119890119904

119894

The optimal order quantity and the promotion level of the

supply chain system-wide can be used as the benchmark forsupply chain coordination to adjust the option parametersThe coordination conditions of supply chain will be given bythe following theorem

Theorem 4 The supply chain of a risk-neutral supplier andtwo risk-averse retailers engaging in promotion competitionwith an option contract and a CVaR criterion can be coordi-nated by the following conditions

1198611= 1198612

1205781

1205782

=1198601

1198602

1 minus119862

119875119894

lt 120578119894⩽ 1

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119885119894⩾ 119875119894

119862 lt 119874119894+ 119864119894lt 119875119894

(25)

In Theorem 4 the first two items show that if the supplychain can be coordinated then the two retailers first achievecompetitive equilibriumThe third item implies that retailersare not very risk-averse that is retailers will pursue profit bytaking some risk LetΔ119889

119894= 119889119894(119890119894+Δ119890119894 1198903minus119894)minus119889119894(119890119894 1198903minus119894)Δ119889119904119894=

119889119894(119890119904

119894+Δ119890119894 119890119904

3minus119894)minus119889119894(119890119904

119894 119890119904

3minus119894) andΔ119889119904

3minus119894= 1198893minus119894(119890119904

3minus119894 119890119904

119894+Δ119890119894)minus

1198893minus119894(119890119904

3minus119894 119890119904

119894) where Δ119890

119894is a small change on the promotion

level of retailer 119894 The fourth item shows that if the promotionlevel of retailer 119894 changes Δ119890

119894units then retailer 119894rsquos demand

will change Δ119889119904119894units and the other retailerrsquos demand will

change Δ1198891199043minus119894

units in the centralized case retailer 119894rsquos demandwill change (119861119904

119894Δ119889119904

119894+119861119904

3minus119894Δ119889119904

3minus119894)119861119894units in the decentralized

case This relationship is brought about by the two retailersrsquocompetition and promotional activities Furthermore thepenalty threshold value must be higher than the sale priceor the supplier will not produce all option order quantityFinally by adjusting the parameters of the option contractthe whole supply chain profit can reach the optimum andthe profits of supply chain members can achieve Paretooptimum

Mathematical Problems in Engineering 7

Table 1 Comparison of the coordination conditions of several supply chains

SC1

SC2

119878119862119894

1198781198623119894

119885 ⩾ 119875 119885 ⩾ 119875 119885119894⩾ 119875119894

119885119894⩾ 119875119894

119875 = 119864 +119875119874

119862119875 = 119864 +

119875119874120578

119862 minus 119875(1 minus 120578)119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119875119894= 119864119894+119875119894119874119894

119862

120578 gt 1 minus119862

119875120578119894gt 1 minus

119862

119875119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

S

S1

S2

R1

PromotionCompetitionR2

C1

C2

SC1

SC

SC2

Figure 2 The relationship of 119878119862 agents

Then we will discuss the subchain coordination condi-tions Figure 2 shows that the supply chain contains twosubchains and the coordination conditions of subchain willbe given by the following theorem

Theorem 5 The subchain (119878119862119894) can be coordinated by the

following conditions

120578119894gt 1 minus

119862

119875119894

119885119894⩾ 119875119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

gt 1

(26)

FromTheorem 5 we can see that the first three coordina-tion conditions of 119878119862

119894are similar to that of the entity supply

chain The last condition implies that Δ119889119894gt Δ119889

119904

119894 which

means that the level of the promotion activity of retailer 119894 indecentralized decision is higher than in centralized decision

Furthermore we can derive the coordination conditionsof supply chain with a neutral supplier and a neutral retailer(1198781198621) supply chain with a neutral supplier and a risk retailer

(1198781198622) and subchain of supply chain with a neutral supplier

and two neutral retailers engaging in promotion competition(1198781198623119894) In the coordination conditions of 119878119862

119894 when 120578

119894and 119890119894

are 1 and 0 respectively we get the coordination conditionsof 1198781198623119894and 119878119862

2 which are consistent with the conclusion of

[39] and when 120578119894= 1 and 119890

119894= 0 the coordination conditions

of 1198781198621are obtained Now we compare them in Table 1 From

Table 1 we find that the penalty cost is higher than theretail price for four supply chains or subchains coordination

conditions Obviously this condition is beneficial for theretailers to fully exercise the option and the supplier hasto produce all option order quantity to reduce loss Table 1shows that the sale price of 119878119862

1is higher than 119878119862

2 which

embodies the characteristic of the retailerrsquos risk aversionSimilar conclusions exist in 119878119862

119894and 119878119862

3119894

6 Numerical Analysis

In this section we carry out numerical experiments underthe model assumption to illustrate our findings We let 119875

1=

55 1198641= 335 119874

1= 58 119875

2= 547 119864

2= 326 119874

2=

67 1205781= 068 120578

2= 071 119862 = 275 and 119885

1= 60

and 1198852= 56 For simplicity we assume that the random

demand variable of each retailer is uniformly distributedon [0 300] and 119889(119890

119894 1198903minus119894) = 100 + 5119890

119894minus 043 sdot 5119890

3minus119894 The

retailersrsquo optimal option order policy and supplierrsquos optimalproduction decision in decentralized case and the optimalsupply chain decision system-wide are shown in Table 2 (notethat the data are rounded) In Table 2 retailersrsquo optimal orderquantity is the same as the supplierrsquos production decisionand the level of two retailersrsquo promotional activity remainsconsistent in decentralized case Furthermore the optimalprofit of supply chain system is the same as that of theretailers and the supplier in decentralized case which impliesthat the supply chain consisting of a risk-neutral supplierand two risk-averse retailers in competition and engagedin promotion is coordinated under the option contract andCVaR criterion

Thenwe analyze the impact of the level of the promotionalactivity on the retailersrsquo order quantities by fixing 119890lowast

1= 85

and 119890lowast2= 85 respectively and varying 119890

2and 1198901from 0 to

200 in steps of 5 corresponding 119890lowast1and 119890lowast2 Figure 3(a) shows

that retailer 1rsquos order quantity will increase when 1198901increases

however retailer 2rsquos order quantity will decrease similar toFigure 3(b) Figures 3(a) and 3(b) also illustrate the uniqueequilibrium point between two risk-averse retailers which isin agreement with the conclusion of Theorem 3

7 Conclusion

This paper investigates an option contract for coordinatinga supply chain with one risk-neutral supplier and two risk-averse retailers engaged in promotion competition Based onthe option contract the optimal option order quantity andthe promotion level of two retailers are obtained with CVaR

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

e1

e2

qlowast

2(e

lowast

2 e1)

qlowast

1(e

lowast

1 e2)

Figure 1 Impact of the promotion level on equilibrium orderquantity

price and exercise price and degree of risk aversion of thetwo retailers is the same)

4 Supplierrsquos Optimal Production Decision

Before the production season the supplier will determinethe production quantity in accordance with the retailersrsquooption order quantity Considering that the two retailers willnot exercise all their options at the beginning of the sellingseason the supplier will reduce production by running therisk of being punished The supplierrsquos profit denoted by 120587119898is

120587119898=

2

sum

119894=1

120587119898

119894=

2

sum

119894=1

119874119894119902lowast

119894+ 119864119894min (119902lowast

119894 119863119894) minus 119862119902

119898

119894

minus 119885119894[min (119902lowast

119894 119863119894) minus 119902119898

119894]+

(12)

where 120587119898

119894is the profit from the supplierrsquos selling of the

production to retailer 119894 and 119902119898

119894is the supplierrsquos option

production for retailer 119894 On the right-hand side of the sumin (12) the first term is the option cost that retailer 119894 paysto the supplier the second term is the supplierrsquos revenuewhen retailers exercise their options the third term is theproduction cost and the last term is the shortage penalty costThe corresponding expected profit is

119864 [120587119898] =

2

sum

119894=1

119864 [120587119898

119894] =

2

sum

119894=1

[(119874119894+ 119864119894) 119902lowast

119894minus 119862119902119898

119894

minus 119864119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

minus 119885119894int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909]

(13)

Property 3 119864[120587119898] is a second-order differentiable function in119902119898

119894and satisfies the following properties

120597119864 [120587119898

119894]

120597119874119894

gt 0

120597119864 [120587119898

119894]

120597119864119894

gt 0

120597119864 [120587119898

119894]

120597119862lt 0

120597119864 [120587119898

119894]

120597119885119894

lt 0

(14)

Property 3 shows that the higher the option price and theexercise price are the higher 119864[120587119898

119894] will be and the higher

the production cost and the penalty cost are the lower 119864[120587119898119894]

will beThese results are consistent with our intuition and theactual situation

In the proof of Property 3 letting the first-order partialderivative of 119864[120587119898

119894] with respect to 119902119898

119894be equal to zero

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 (119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)) = 0 (15)

According to (15) the supplierrsquos production decision forretailer 119894 denoted as 119902119898Δ

119894 is given by

119902119898Δ

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866119898

119894) (16)

where 119866119898119894= (119885119894minus 119862)119885

119894

The first-order partial derivative of 119902119898Δ119894

with respect to119885119894

is 120597119902119898Δ119894120597119885119894= 119862119885

2

119894119891(119902119898Δ

119894minus 119889119894(119890lowast

119894 1198903minus119894)) gt 0 which implies

that for given 119862 119890lowast119894 and 119890

3minus119894 the supplierrsquos production is

increasing in 119885119894 In other words the supplier will choose to

produce more options to reduce the loss when the shortagepenalty cost increases But if the shortage penalty cost isgreater than a certain critical value the supplier will produceall the options or his marginal loss will be greater than themarginal profit Combining (7) and (15) yields the shortagepenalty threshold value

119885119894=

119860119894119862

119860119894minus 119861119894120578119894

(17)

The first-order partial derivative of 119885119894with respect to 120578

119894is

120597119885119894120597120578119894= 119860119894119861119894119862(119860119894minus 119861119894120578119894)2gt 0 which indicates that

the penalty threshold value 119885119894will be higher when retailer

119894 is not very risk-averse According to the above analysisit is clear that the penalty threshold value directly impactsthe supplierrsquos production decision The supplierrsquos optimalproduction decision for retailer 119894 is given by

119902119898lowast

119894=

119902119898Δ

119894 119885119894lt 119885119894

119902lowast

119894 119885

119894⩾ 119885119894

(18)

From (18) it is known thatwhen119885119894is less than119885

119894 the supplier

will accept the penalty and will produce only 119902119894Δ119898

units of

6 Mathematical Problems in Engineering

options for retailer 119894 otherwise he will produce all optionorder quantity At the same time the total optimal productionquantity 119902119898lowast of the supplier for two retailers is given by

119902119898lowast

=

119902119898Δ

1+ 119902119898Δ

2 1198851lt 1198851 1198852lt 1198852

119902119898Δ

1+ 119902lowast

2 119885

1lt 1198851 1198852⩾ 1198852

119902lowast

1+ 119902119898Δ

2 119885

1⩾ 1198851 1198852lt 1198852

119902lowast

1+ 119902lowast

2 119885

1⩾ 1198851 1198852⩾ 1198852

(19)

When 120578119894= 1 the penalty threshold value is119885

119894= 119860119894119862119874119894

and the corresponding optimal production decision 1199021198980lowast119894

ofthe supplier for retailer 119894 is

1199021198980lowast

119894=

1199021198980Δ

119894 119885119894lt 119885119894

1199020lowast

119894 119885

119894⩾ 119885119894

(20)

Obviously 119885119894lt 119885119894and 119902119898lowast119894

lt 1199021198980lowast

119894

5 Supply Chain Coordination

It is well known that the optimal decision of supply chainsystem-wide is the benchmark for supply chain coordinationTo derive the optimal decision of the channel the supplychain is taken as one entity and the profit of the supply chainsystem is formulated and denoted as 120587119904

120587119904=

2

sum

119894=1

[119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)] (21)

where 119902119904119894and 119890119904119894are respectively the order quantity and the

promotion level for retailer 119894 in the supply chain system Onthe right-hand side of the sum in (21) the first term is thesales revenue the second term is the production cost and thelast term is the promotion cost The corresponding expectedprofit is

119864 [120587119904] =

2

sum

119894=1

119864 [119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)]

=

2

sum

119894=1

[minus119875119894int

119902119904

119894minus119889119904

119894

0

119865 (119909) 119889119909 + (119875119894 minus 119862) 119902119904

119894minus 119892119894(119890119904

119894)]

(22)

Obviously the optimal decision of the supply chain canbe obtained by maximizing (22) Let the first-order partialderivative of 119864[120587119904] with respect to 119902119904

119894and 119890119904119894be equal to zero

120597119864 [120587119904]

120597119902119904

119894

= 119861119904

119894minus 119875119894119865 (119862119904

119894) = 0

120597119864 [120587119904]

120597119890119904

119894

= minus119866119894(119890119904

119894) + 119875119894119865 (119862119904

119894)119898119904

119894119894

+ 1198753minus119894119865 (119862119904

3minus119894)119898119904

3minus119894119894= 0

(23)

where 119862119904119894= 119902119904

119894minus 119889119904

119894 119894 = 1 2 In addition the leading

principleminors ofmatrix of119864[120587119904] are as followsminus119875119894119891(119862119904

119894) lt

0 119875119894119891(119862119904

119894)119867119894+ 1198753minus119894119891(119862119904

119894)119891(119862119904

3minus119894)(1198983minus119894119894

)2gt 0 The optimal

decision for the channel in (22) can be obtained as follows

119902119904lowast

119894= 119889119904lowast

119894+ 119865minus1(119866119904

119894)

119890119904lowast

119894= 119866minus1

119894(119898119904

119894119894119861119904

119894+ 119898119904

3minus119894119894119861119904

3minus119894)

(24)

where 119889119904lowast119894= 119889119894(119890119904lowast

119894 119890119904lowast

3minus119894) 119861119904119894= 119875119894minus 119862 119861119904

3minus119894= 1198753minus119894

minus 119862 119866119904119894=

119861119904

119894119875119894119898119904119894119894= 1198891015840

119894119890119904

119894

and1198981199043minus119894119894

= 1198891015840

3minus119894119890119904

119894

The optimal order quantity and the promotion level of the

supply chain system-wide can be used as the benchmark forsupply chain coordination to adjust the option parametersThe coordination conditions of supply chain will be given bythe following theorem

Theorem 4 The supply chain of a risk-neutral supplier andtwo risk-averse retailers engaging in promotion competitionwith an option contract and a CVaR criterion can be coordi-nated by the following conditions

1198611= 1198612

1205781

1205782

=1198601

1198602

1 minus119862

119875119894

lt 120578119894⩽ 1

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119885119894⩾ 119875119894

119862 lt 119874119894+ 119864119894lt 119875119894

(25)

In Theorem 4 the first two items show that if the supplychain can be coordinated then the two retailers first achievecompetitive equilibriumThe third item implies that retailersare not very risk-averse that is retailers will pursue profit bytaking some risk LetΔ119889

119894= 119889119894(119890119894+Δ119890119894 1198903minus119894)minus119889119894(119890119894 1198903minus119894)Δ119889119904119894=

119889119894(119890119904

119894+Δ119890119894 119890119904

3minus119894)minus119889119894(119890119904

119894 119890119904

3minus119894) andΔ119889119904

3minus119894= 1198893minus119894(119890119904

3minus119894 119890119904

119894+Δ119890119894)minus

1198893minus119894(119890119904

3minus119894 119890119904

119894) where Δ119890

119894is a small change on the promotion

level of retailer 119894 The fourth item shows that if the promotionlevel of retailer 119894 changes Δ119890

119894units then retailer 119894rsquos demand

will change Δ119889119904119894units and the other retailerrsquos demand will

change Δ1198891199043minus119894

units in the centralized case retailer 119894rsquos demandwill change (119861119904

119894Δ119889119904

119894+119861119904

3minus119894Δ119889119904

3minus119894)119861119894units in the decentralized

case This relationship is brought about by the two retailersrsquocompetition and promotional activities Furthermore thepenalty threshold value must be higher than the sale priceor the supplier will not produce all option order quantityFinally by adjusting the parameters of the option contractthe whole supply chain profit can reach the optimum andthe profits of supply chain members can achieve Paretooptimum

Mathematical Problems in Engineering 7

Table 1 Comparison of the coordination conditions of several supply chains

SC1

SC2

119878119862119894

1198781198623119894

119885 ⩾ 119875 119885 ⩾ 119875 119885119894⩾ 119875119894

119885119894⩾ 119875119894

119875 = 119864 +119875119874

119862119875 = 119864 +

119875119874120578

119862 minus 119875(1 minus 120578)119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119875119894= 119864119894+119875119894119874119894

119862

120578 gt 1 minus119862

119875120578119894gt 1 minus

119862

119875119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

S

S1

S2

R1

PromotionCompetitionR2

C1

C2

SC1

SC

SC2

Figure 2 The relationship of 119878119862 agents

Then we will discuss the subchain coordination condi-tions Figure 2 shows that the supply chain contains twosubchains and the coordination conditions of subchain willbe given by the following theorem

Theorem 5 The subchain (119878119862119894) can be coordinated by the

following conditions

120578119894gt 1 minus

119862

119875119894

119885119894⩾ 119875119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

gt 1

(26)

FromTheorem 5 we can see that the first three coordina-tion conditions of 119878119862

119894are similar to that of the entity supply

chain The last condition implies that Δ119889119894gt Δ119889

119904

119894 which

means that the level of the promotion activity of retailer 119894 indecentralized decision is higher than in centralized decision

Furthermore we can derive the coordination conditionsof supply chain with a neutral supplier and a neutral retailer(1198781198621) supply chain with a neutral supplier and a risk retailer

(1198781198622) and subchain of supply chain with a neutral supplier

and two neutral retailers engaging in promotion competition(1198781198623119894) In the coordination conditions of 119878119862

119894 when 120578

119894and 119890119894

are 1 and 0 respectively we get the coordination conditionsof 1198781198623119894and 119878119862

2 which are consistent with the conclusion of

[39] and when 120578119894= 1 and 119890

119894= 0 the coordination conditions

of 1198781198621are obtained Now we compare them in Table 1 From

Table 1 we find that the penalty cost is higher than theretail price for four supply chains or subchains coordination

conditions Obviously this condition is beneficial for theretailers to fully exercise the option and the supplier hasto produce all option order quantity to reduce loss Table 1shows that the sale price of 119878119862

1is higher than 119878119862

2 which

embodies the characteristic of the retailerrsquos risk aversionSimilar conclusions exist in 119878119862

119894and 119878119862

3119894

6 Numerical Analysis

In this section we carry out numerical experiments underthe model assumption to illustrate our findings We let 119875

1=

55 1198641= 335 119874

1= 58 119875

2= 547 119864

2= 326 119874

2=

67 1205781= 068 120578

2= 071 119862 = 275 and 119885

1= 60

and 1198852= 56 For simplicity we assume that the random

demand variable of each retailer is uniformly distributedon [0 300] and 119889(119890

119894 1198903minus119894) = 100 + 5119890

119894minus 043 sdot 5119890

3minus119894 The

retailersrsquo optimal option order policy and supplierrsquos optimalproduction decision in decentralized case and the optimalsupply chain decision system-wide are shown in Table 2 (notethat the data are rounded) In Table 2 retailersrsquo optimal orderquantity is the same as the supplierrsquos production decisionand the level of two retailersrsquo promotional activity remainsconsistent in decentralized case Furthermore the optimalprofit of supply chain system is the same as that of theretailers and the supplier in decentralized case which impliesthat the supply chain consisting of a risk-neutral supplierand two risk-averse retailers in competition and engagedin promotion is coordinated under the option contract andCVaR criterion

Thenwe analyze the impact of the level of the promotionalactivity on the retailersrsquo order quantities by fixing 119890lowast

1= 85

and 119890lowast2= 85 respectively and varying 119890

2and 1198901from 0 to

200 in steps of 5 corresponding 119890lowast1and 119890lowast2 Figure 3(a) shows

that retailer 1rsquos order quantity will increase when 1198901increases

however retailer 2rsquos order quantity will decrease similar toFigure 3(b) Figures 3(a) and 3(b) also illustrate the uniqueequilibrium point between two risk-averse retailers which isin agreement with the conclusion of Theorem 3

7 Conclusion

This paper investigates an option contract for coordinatinga supply chain with one risk-neutral supplier and two risk-averse retailers engaged in promotion competition Based onthe option contract the optimal option order quantity andthe promotion level of two retailers are obtained with CVaR

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

options for retailer 119894 otherwise he will produce all optionorder quantity At the same time the total optimal productionquantity 119902119898lowast of the supplier for two retailers is given by

119902119898lowast

=

119902119898Δ

1+ 119902119898Δ

2 1198851lt 1198851 1198852lt 1198852

119902119898Δ

1+ 119902lowast

2 119885

1lt 1198851 1198852⩾ 1198852

119902lowast

1+ 119902119898Δ

2 119885

1⩾ 1198851 1198852lt 1198852

119902lowast

1+ 119902lowast

2 119885

1⩾ 1198851 1198852⩾ 1198852

(19)

When 120578119894= 1 the penalty threshold value is119885

119894= 119860119894119862119874119894

and the corresponding optimal production decision 1199021198980lowast119894

ofthe supplier for retailer 119894 is

1199021198980lowast

119894=

1199021198980Δ

119894 119885119894lt 119885119894

1199020lowast

119894 119885

119894⩾ 119885119894

(20)

Obviously 119885119894lt 119885119894and 119902119898lowast119894

lt 1199021198980lowast

119894

5 Supply Chain Coordination

It is well known that the optimal decision of supply chainsystem-wide is the benchmark for supply chain coordinationTo derive the optimal decision of the channel the supplychain is taken as one entity and the profit of the supply chainsystem is formulated and denoted as 120587119904

120587119904=

2

sum

119894=1

[119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)] (21)

where 119902119904119894and 119890119904119894are respectively the order quantity and the

promotion level for retailer 119894 in the supply chain system Onthe right-hand side of the sum in (21) the first term is thesales revenue the second term is the production cost and thelast term is the promotion cost The corresponding expectedprofit is

119864 [120587119904] =

2

sum

119894=1

119864 [119875119894min (119902119904

119894 119863119894) minus 119862119902

119904

119894minus 119892119894(119890119904

119894)]

=

2

sum

119894=1

[minus119875119894int

119902119904

119894minus119889119904

119894

0

119865 (119909) 119889119909 + (119875119894 minus 119862) 119902119904

119894minus 119892119894(119890119904

119894)]

(22)

Obviously the optimal decision of the supply chain canbe obtained by maximizing (22) Let the first-order partialderivative of 119864[120587119904] with respect to 119902119904

119894and 119890119904119894be equal to zero

120597119864 [120587119904]

120597119902119904

119894

= 119861119904

119894minus 119875119894119865 (119862119904

119894) = 0

120597119864 [120587119904]

120597119890119904

119894

= minus119866119894(119890119904

119894) + 119875119894119865 (119862119904

119894)119898119904

119894119894

+ 1198753minus119894119865 (119862119904

3minus119894)119898119904

3minus119894119894= 0

(23)

where 119862119904119894= 119902119904

119894minus 119889119904

119894 119894 = 1 2 In addition the leading

principleminors ofmatrix of119864[120587119904] are as followsminus119875119894119891(119862119904

119894) lt

0 119875119894119891(119862119904

119894)119867119894+ 1198753minus119894119891(119862119904

119894)119891(119862119904

3minus119894)(1198983minus119894119894

)2gt 0 The optimal

decision for the channel in (22) can be obtained as follows

119902119904lowast

119894= 119889119904lowast

119894+ 119865minus1(119866119904

119894)

119890119904lowast

119894= 119866minus1

119894(119898119904

119894119894119861119904

119894+ 119898119904

3minus119894119894119861119904

3minus119894)

(24)

where 119889119904lowast119894= 119889119894(119890119904lowast

119894 119890119904lowast

3minus119894) 119861119904119894= 119875119894minus 119862 119861119904

3minus119894= 1198753minus119894

minus 119862 119866119904119894=

119861119904

119894119875119894119898119904119894119894= 1198891015840

119894119890119904

119894

and1198981199043minus119894119894

= 1198891015840

3minus119894119890119904

119894

The optimal order quantity and the promotion level of the

supply chain system-wide can be used as the benchmark forsupply chain coordination to adjust the option parametersThe coordination conditions of supply chain will be given bythe following theorem

Theorem 4 The supply chain of a risk-neutral supplier andtwo risk-averse retailers engaging in promotion competitionwith an option contract and a CVaR criterion can be coordi-nated by the following conditions

1198611= 1198612

1205781

1205782

=1198601

1198602

1 minus119862

119875119894

lt 120578119894⩽ 1

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119885119894⩾ 119875119894

119862 lt 119874119894+ 119864119894lt 119875119894

(25)

In Theorem 4 the first two items show that if the supplychain can be coordinated then the two retailers first achievecompetitive equilibriumThe third item implies that retailersare not very risk-averse that is retailers will pursue profit bytaking some risk LetΔ119889

119894= 119889119894(119890119894+Δ119890119894 1198903minus119894)minus119889119894(119890119894 1198903minus119894)Δ119889119904119894=

119889119894(119890119904

119894+Δ119890119894 119890119904

3minus119894)minus119889119894(119890119904

119894 119890119904

3minus119894) andΔ119889119904

3minus119894= 1198893minus119894(119890119904

3minus119894 119890119904

119894+Δ119890119894)minus

1198893minus119894(119890119904

3minus119894 119890119904

119894) where Δ119890

119894is a small change on the promotion

level of retailer 119894 The fourth item shows that if the promotionlevel of retailer 119894 changes Δ119890

119894units then retailer 119894rsquos demand

will change Δ119889119904119894units and the other retailerrsquos demand will

change Δ1198891199043minus119894

units in the centralized case retailer 119894rsquos demandwill change (119861119904

119894Δ119889119904

119894+119861119904

3minus119894Δ119889119904

3minus119894)119861119894units in the decentralized

case This relationship is brought about by the two retailersrsquocompetition and promotional activities Furthermore thepenalty threshold value must be higher than the sale priceor the supplier will not produce all option order quantityFinally by adjusting the parameters of the option contractthe whole supply chain profit can reach the optimum andthe profits of supply chain members can achieve Paretooptimum

Mathematical Problems in Engineering 7

Table 1 Comparison of the coordination conditions of several supply chains

SC1

SC2

119878119862119894

1198781198623119894

119885 ⩾ 119875 119885 ⩾ 119875 119885119894⩾ 119875119894

119885119894⩾ 119875119894

119875 = 119864 +119875119874

119862119875 = 119864 +

119875119874120578

119862 minus 119875(1 minus 120578)119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119875119894= 119864119894+119875119894119874119894

119862

120578 gt 1 minus119862

119875120578119894gt 1 minus

119862

119875119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

S

S1

S2

R1

PromotionCompetitionR2

C1

C2

SC1

SC

SC2

Figure 2 The relationship of 119878119862 agents

Then we will discuss the subchain coordination condi-tions Figure 2 shows that the supply chain contains twosubchains and the coordination conditions of subchain willbe given by the following theorem

Theorem 5 The subchain (119878119862119894) can be coordinated by the

following conditions

120578119894gt 1 minus

119862

119875119894

119885119894⩾ 119875119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

gt 1

(26)

FromTheorem 5 we can see that the first three coordina-tion conditions of 119878119862

119894are similar to that of the entity supply

chain The last condition implies that Δ119889119894gt Δ119889

119904

119894 which

means that the level of the promotion activity of retailer 119894 indecentralized decision is higher than in centralized decision

Furthermore we can derive the coordination conditionsof supply chain with a neutral supplier and a neutral retailer(1198781198621) supply chain with a neutral supplier and a risk retailer

(1198781198622) and subchain of supply chain with a neutral supplier

and two neutral retailers engaging in promotion competition(1198781198623119894) In the coordination conditions of 119878119862

119894 when 120578

119894and 119890119894

are 1 and 0 respectively we get the coordination conditionsof 1198781198623119894and 119878119862

2 which are consistent with the conclusion of

[39] and when 120578119894= 1 and 119890

119894= 0 the coordination conditions

of 1198781198621are obtained Now we compare them in Table 1 From

Table 1 we find that the penalty cost is higher than theretail price for four supply chains or subchains coordination

conditions Obviously this condition is beneficial for theretailers to fully exercise the option and the supplier hasto produce all option order quantity to reduce loss Table 1shows that the sale price of 119878119862

1is higher than 119878119862

2 which

embodies the characteristic of the retailerrsquos risk aversionSimilar conclusions exist in 119878119862

119894and 119878119862

3119894

6 Numerical Analysis

In this section we carry out numerical experiments underthe model assumption to illustrate our findings We let 119875

1=

55 1198641= 335 119874

1= 58 119875

2= 547 119864

2= 326 119874

2=

67 1205781= 068 120578

2= 071 119862 = 275 and 119885

1= 60

and 1198852= 56 For simplicity we assume that the random

demand variable of each retailer is uniformly distributedon [0 300] and 119889(119890

119894 1198903minus119894) = 100 + 5119890

119894minus 043 sdot 5119890

3minus119894 The

retailersrsquo optimal option order policy and supplierrsquos optimalproduction decision in decentralized case and the optimalsupply chain decision system-wide are shown in Table 2 (notethat the data are rounded) In Table 2 retailersrsquo optimal orderquantity is the same as the supplierrsquos production decisionand the level of two retailersrsquo promotional activity remainsconsistent in decentralized case Furthermore the optimalprofit of supply chain system is the same as that of theretailers and the supplier in decentralized case which impliesthat the supply chain consisting of a risk-neutral supplierand two risk-averse retailers in competition and engagedin promotion is coordinated under the option contract andCVaR criterion

Thenwe analyze the impact of the level of the promotionalactivity on the retailersrsquo order quantities by fixing 119890lowast

1= 85

and 119890lowast2= 85 respectively and varying 119890

2and 1198901from 0 to

200 in steps of 5 corresponding 119890lowast1and 119890lowast2 Figure 3(a) shows

that retailer 1rsquos order quantity will increase when 1198901increases

however retailer 2rsquos order quantity will decrease similar toFigure 3(b) Figures 3(a) and 3(b) also illustrate the uniqueequilibrium point between two risk-averse retailers which isin agreement with the conclusion of Theorem 3

7 Conclusion

This paper investigates an option contract for coordinatinga supply chain with one risk-neutral supplier and two risk-averse retailers engaged in promotion competition Based onthe option contract the optimal option order quantity andthe promotion level of two retailers are obtained with CVaR

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Comparison of the coordination conditions of several supply chains

SC1

SC2

119878119862119894

1198781198623119894

119885 ⩾ 119875 119885 ⩾ 119875 119885119894⩾ 119875119894

119885119894⩾ 119875119894

119875 = 119864 +119875119874

119862119875 = 119864 +

119875119874120578

119862 minus 119875(1 minus 120578)119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119875119894= 119864119894+119875119894119874119894

119862

120578 gt 1 minus119862

119875120578119894gt 1 minus

119862

119875119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

S

S1

S2

R1

PromotionCompetitionR2

C1

C2

SC1

SC

SC2

Figure 2 The relationship of 119878119862 agents

Then we will discuss the subchain coordination condi-tions Figure 2 shows that the supply chain contains twosubchains and the coordination conditions of subchain willbe given by the following theorem

Theorem 5 The subchain (119878119862119894) can be coordinated by the

following conditions

120578119894gt 1 minus

119862

119875119894

119885119894⩾ 119875119894

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894

119898119904

119894119894

=119861119904

119894

119861119894

gt 1

(26)

FromTheorem 5 we can see that the first three coordina-tion conditions of 119878119862

119894are similar to that of the entity supply

chain The last condition implies that Δ119889119894gt Δ119889

119904

119894 which

means that the level of the promotion activity of retailer 119894 indecentralized decision is higher than in centralized decision

Furthermore we can derive the coordination conditionsof supply chain with a neutral supplier and a neutral retailer(1198781198621) supply chain with a neutral supplier and a risk retailer

(1198781198622) and subchain of supply chain with a neutral supplier

and two neutral retailers engaging in promotion competition(1198781198623119894) In the coordination conditions of 119878119862

119894 when 120578

119894and 119890119894

are 1 and 0 respectively we get the coordination conditionsof 1198781198623119894and 119878119862

2 which are consistent with the conclusion of

[39] and when 120578119894= 1 and 119890

119894= 0 the coordination conditions

of 1198781198621are obtained Now we compare them in Table 1 From

Table 1 we find that the penalty cost is higher than theretail price for four supply chains or subchains coordination

conditions Obviously this condition is beneficial for theretailers to fully exercise the option and the supplier hasto produce all option order quantity to reduce loss Table 1shows that the sale price of 119878119862

1is higher than 119878119862

2 which

embodies the characteristic of the retailerrsquos risk aversionSimilar conclusions exist in 119878119862

119894and 119878119862

3119894

6 Numerical Analysis

In this section we carry out numerical experiments underthe model assumption to illustrate our findings We let 119875

1=

55 1198641= 335 119874

1= 58 119875

2= 547 119864

2= 326 119874

2=

67 1205781= 068 120578

2= 071 119862 = 275 and 119885

1= 60

and 1198852= 56 For simplicity we assume that the random

demand variable of each retailer is uniformly distributedon [0 300] and 119889(119890

119894 1198903minus119894) = 100 + 5119890

119894minus 043 sdot 5119890

3minus119894 The

retailersrsquo optimal option order policy and supplierrsquos optimalproduction decision in decentralized case and the optimalsupply chain decision system-wide are shown in Table 2 (notethat the data are rounded) In Table 2 retailersrsquo optimal orderquantity is the same as the supplierrsquos production decisionand the level of two retailersrsquo promotional activity remainsconsistent in decentralized case Furthermore the optimalprofit of supply chain system is the same as that of theretailers and the supplier in decentralized case which impliesthat the supply chain consisting of a risk-neutral supplierand two risk-averse retailers in competition and engagedin promotion is coordinated under the option contract andCVaR criterion

Thenwe analyze the impact of the level of the promotionalactivity on the retailersrsquo order quantities by fixing 119890lowast

1= 85

and 119890lowast2= 85 respectively and varying 119890

2and 1198901from 0 to

200 in steps of 5 corresponding 119890lowast1and 119890lowast2 Figure 3(a) shows

that retailer 1rsquos order quantity will increase when 1198901increases

however retailer 2rsquos order quantity will decrease similar toFigure 3(b) Figures 3(a) and 3(b) also illustrate the uniqueequilibrium point between two risk-averse retailers which isin agreement with the conclusion of Theorem 3

7 Conclusion

This paper investigates an option contract for coordinatinga supply chain with one risk-neutral supplier and two risk-averse retailers engaged in promotion competition Based onthe option contract the optimal option order quantity andthe promotion level of two retailers are obtained with CVaR

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 Results on optimal decision in decentralized case and centralized case

Decentralized caseRetailer 119890

lowast

1119890lowast

2119902119903lowast

1119902119903lowast

2119902119903lowast

119864120587119903lowast

1119864120587119903lowast

2119864120587119903lowast CVaRlowast

1CVaRlowast

2

85 85 507 507 114 3931 3931 7862 1432 1432

Supplier mdash mdash 119902119898lowast

1119902119898lowast

2119902119898lowast

119864120587119898lowast

1119864120587119898lowast

2119864120587119898lowast mdash mdash

mdash mdash 507 507 114 5350 5350 10700 mdash mdash

Centralized case Supply chain 119890119904lowast

1119890119904lowast

2119902119904lowast

1119902119904lowast

2119902119904lowast

119864120587119904lowast

1119864120587119904lowast

2119864120587119904lowast mdash mdash

85 85 507 507 114 9281 9281 18562 mdash mdash

Impact of e1 on order quantity in decentralized case

e1

0 20 40 60 80 100 120 140 160 180 2000

200

400

600q

800

1000

1200

(85 507)

q1

q2

(a)

0 20 40 60 80 100 120 140 160 180 200

e2

Impact of e2 on order quantity in decentralized case

0

200

400

600q

800

1000

1200

(85 507)

q1

q2

(b)

Figure 3 Impact of level of promotional activity on order quantities in decentralized case

criterion The impact of the promotion level on the optimalorder quantity of each retailer is studied and a unique Nashequilibrium between two retailers is derived Based on theretailersrsquo optimal option order policy the supplierrsquos opti-mal production decision is further obtained by maximizingexpected profit Furthermore we discuss the coordinationissues of the supply chain system and its subchain and givethe corresponding coordination conditions Both in supplychain and in its subchain the penalty cost threshold valueshould be higher than selling price to stimulate the supplier toproduce all option order quantity and the retailersrsquo degree ofrisk aversion should not be too high Numerical experimentsillustrate the unique Nash equilibrium between two retailersand show that the optimal order quantity of each retailerincreases (decreases) with its own (competitorrsquos) promotionlevel

Of course this study includes some limitations whichrequire further exploration in the future For example thesupply chain that we studied above is assumed to have arisk-neutral supplier which implies that the supplier hasno risk preference However it is known that the supplierrsquosrisk attitude determines the option price and the exerciseprice which will in turn affect the option order quantityand the coordination conditions Therefore in future workthe supply chain with a risk-averse supplier and risk-averseretailers can be taken into account

Appendix

Proof of Property 1 The first-order and second-order partialderivatives of 119880

119894in (6) with respect to 120572 are as follows

120597119880119894

120597120572=

1 120572119894⩽ 119882119894

1 minus1

120578119894

119865(120572 minus119882

119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

1 minus1

120578119894

120572119894gt 119881119894

(A1)

1205972119880119894

1205971205722=

minus1

120578119894119860119894

119891(120572119894minus119882119894

119860119894

) 119882119894lt 120572 ⩽ 119881

119894

0 others(A2)

Obviously 12059721198801198941205971205722⩽ 0 which implies that 119880

119894is a differen-

tiable concave function of 120572 The stationary point denoted as120572lowast(119902119894 119890119894) is the maximum point From (A1)

120572lowast(119902119894 119890119894) =

119881119894 119862

119894lt 119865minus1(120572)

119882119894+ 119860119894119865minus1(120578119894) 119862

119894⩾ 119865minus1(120572)

(A3)

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

When 119862119894⩾ 119865minus1(120572) then 119882

119894lt 120572lowast(119902119894 119890119894) = 119882

119894+

119860119894119865minus1(120578119894) ⩽ 119881119894 and

119862120578119894[120587119903

119894] = 119882

119894+ 119860119894119865minus1(120578119894) minus

1

120578119894

sdot int

(120572lowast(119902119894 119890119894)minus119882119894)119860119894

0

(120572lowast(119902119894 119890119894) minus 119882

119894minus 119860119894119909)

sdot 119891 (119909) 119889119909 = 119882119894 + 119860 119894 [119865minus1(120578119894) minus

1

120578119894

sdot int

119865minus1(120578119894)

0

119865 (119909) 119889119909]

(A4)

However 120597119862120578119894[120587119903

119894]120597119902119894= minus119874

119894lt 0 and therefore the

maximum point of 119862120578119894[120587119903

119894] does not exist in this area

When 119862119894lt 119865minus1(120572119894) then 120572lowast

119894(119902119894 119890119894) = 119881119894 and

119862120578119894[120587119903

119894] = 119881119894

minus1

120578119894

int

(119881119894minus119882119894)119860119894

0

(119881119894minus119882119894minus 119860119894119909)119891 (119909) 119889119909

= 119881119894minus1

120578119894

int

119862119894

0

119860119894119865 (119909) 119889119909

(A5)

In this case the Hessian matrix of 119862120578119894[120587119903

119894] on (119902

119894 119890119894) is

[[[

[

minus119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894)

119898119894119894119860119894

120578119894

119891 (119862119894) minus119867

119894minus(119898119894119894)2119860119894

120578119894

119891 (119862119894)

]]]

]

(A6)

The leading principle minors of the Hessian matrix are asfollows minus(119860

119894120578119894)119891(119862119894) lt 0 and (119867

119894119860119894120578119894)119891(119862119894) gt 0

which implies that the Hessianmatrix is strictly negative andtherefore 119862

120578119894[120587119903

119894] is a strictly differentiable concave function

on (119902119894 119890119894)

Proof of Theorem 2 From Property 1 it is known that thefirst-order partial derivative of 119862

120578119894[120587119903

119894] with respect to 119902

119894and

119890119894satisfies the following conditions

120597119862120578119894[120587119903

119894]

120597119902119894

= 119861119894minus119860119894

120578119894

119865 (119862119894) = 0

120597119862120578119894[120587119903

119894]

120597119890119894

= minus119866119894(119890119894) +

119898119894119894119860119894

120578119894

119865 (119862119894) = 0

(A7)

With some algebra retailer 119894rsquos optimal option ordering policycan be obtained as follows

119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894)

119890lowast

119894= 119866minus1

119894(119898119894119894119861119894)

(A8)

Substituting (119902lowast

119894 119890lowast

119894) into (6) leads to the following

119862lowast

120578119894[120587119903

119894(119902lowast

119894 119890lowast

119894)] = (119860

119894120578119894) int119866120578119894

0119865minus1(119905)119889119905+119861

119894119889119894(119890lowast

119894 1198903minus119894)minus119892119894(119890lowast

119894)

where 119866120578119894= (119861119894119860119894)120578119894119898119894119894= 1198891015840

119894119890119894(119890119894 1198903minus119894) and 119905 = 119865(119909)

Proof of Property 2 According to (A8) it is known that119902lowast

119894= 119889119894(119890lowast

119894 1198903minus119894) + 119865minus1(119866120578119894) With some algebra it can be

determined that ((119875119894minus119864119894minus119874119894)(119875119894minus119864119894))120578119894= 119865(119902

lowast

119894minus119889119894(119890lowast

119894 1198903minus119894))

Using the chain rule for the derivative and 119891(119862119894) gt 0 the

partial derivative of 119902lowast119894on 120578119894 119874119894 and 119864

119894can be obtained as

follows

120597119902lowast

119894

120597119874119894

= minus120578119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597119864119894

= minus119874119894120578119894

(119860119894)2sdot

1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]lt 0

120597119902lowast

119894

120597120578119894

=119861119894

119860119894

sdot1

119891 [119902lowast

119894minus 119889119894(119890lowast

119894 1198903minus119894)]gt 0

(A9)

According to (A8) and the assumption in Section 3120597119902lowast

1198941205971198903minus119894

= 119889[119889119894(119890lowast

119894 1198903minus119894)]1198891198903minus119894

lt 0 The desired resultfollows and the proof is complete

Proof ofTheorem 3 From Property 1 it is known that 119862120578119894[120587119903

119894]

is strictly concave on (119902119894 119890119894) and that the strategy space

[0 +infin) times [0 +infin) of retailer 119894 is a compact convex setTherefore there is a pure strategy Nash equilibrium betweenthe two retailers Furthermore the equilibrium strategy ofthe two retailers in the game must be inside the space andtherefore the balance of the game is unique [40]

Proof of Property 3 Note that 119902lowast119894

gt 119902119898

119894gt 0 119891[119902119898

119894minus

119889119894(119890lowast

119894 1198903minus119894)] gt 0 and 0 lt 119865(119909) lt 1 From (13) the first-order

and second-order partial derivatives of 119864[120587119898119894] on 119902119898

119894and the

first-order partial derivative of 119864[120587119898119894] on119874

119894 119864119894119862 and119885

119894can

be obtained as follows

120597119864 [120587119898

119894]

120597119902119898

119894

= minus119862 + 119885119894minus 119885119894119865 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)]

1205972119864 [120587119898

119894]

120597 (119902119898

119894)2= minus119885119894119891 [119902119898

119894minus 119889119894(119890lowast

119894 1198903minus119894)] lt 0

120597119864 [120587119898

119894]

120597119874119894

= 119902lowast

119894gt 0

120597119864 [120587119898

119894]

120597119864119894

= 119889119894(119890lowast

119894 1198903minus119894) + int

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

0

119865 (119909) 119889119909

gt 0

120597119864 [120587119898

119894]

120597119862= minus (119902

119898

1+ 119902119898

2) lt 0

120597119864 [120587119898

119894]

120597119885119894

= minusint

119902lowast

119894minus119889119894(119890lowast

1198941198903minus119894)

119902119898

119894minus119889119894(119890lowast

1198941198903minus119894)

119865 (119909) 119889119909 lt 0

(A10)

The desired result follows and the proof is complete

Proof of Theorem 4 The optimal decision of the supply chainsystem-wide can be used as a benchmark to search for thecoordination conditions First if 119902lowast

119894= 119902119904lowast

119894 then 119890

lowast

119894= 119890119904lowast

119894

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

119889lowast

119894= 119889119904lowast

119894 and 119866

120578119894= 119866119904

119894 that is ((119875

119894minus 119864119894minus 119874119894)(119875119894minus 119864119894))120578119894=

(119875119894minus 119862)119875

119894 Then the following result can be obtained

1 minus119862

119875119894

lt 120578119894⩽ 1

119875119894= 119864119894+

119875119894119874119894120578119894

119862 minus 119875119894(1 minus 120578

119894)

119898119894119894=119861119904

119894119898119904

119894119894+ 119861119904

3minus119894119898119904

3minus119894119894

119861119894

(A11)

Based on the equilibrium competition every retailer willcompete with the supplier at the same time If 119902119898lowast = sum2

119894=1119902119904

119894

then 119902119898lowast

= sum2

119894=1119902119898Δ

119894 or 119902119898lowast = 119902

119898Δ

1+ 119902119898lowast

2 or 119902119898lowast =

119902119898lowast

1+119902119898Δ

2 but 119902119904 lt sum2

119894=1119902lowast

119894 which is in conflict with 119902119904 = 119902lowast

119894

Therefore the supplier must satisfy each retailerrsquos optimalordering quantity that is 119902119898lowast = sum2

119894=1119902lowast

119894 which indicates that

119885119894⩾ 119885119894 Moreover for 119885

119894= 119875119894 119885119894⩾ 119875119894

Proof of Theorem 5 It is similar to the proof of Theorem 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research has been supported by the National NaturalScience Foundation of China under Grant 61273233

References

[1] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[2] D Barnes-Schuster Y Bassok and R Anupindi ldquoCoordinationand flexibility in supply contracts with optionsrdquoManufacturingand Service Operations Management vol 4 no 3 pp 171ndash2072002

[3] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp133ndash140 2008

[4] A A Tsay andW S Lovejoy ldquoQuantity flexibility contracts andsupply chain performancerdquo Manufacturing and Service Opera-tions Management vol 1 no 2 pp 89ndash111 1999

[5] M Lariviere ldquoInducing forecast revelation through restrictedreturns EBOLrdquo 2002 httpbctimwust1educalendarmedi-afilesForecasts 2002pdf

[6] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[7] C L Munson and M J Rosenblatt ldquoCoordinating a three-levelsupply chain with quantity discountsrdquo IIE Transactions vol 33no 5 pp 371ndash384 2001

[8] H Gurnani ldquoA study of quantity discount pricing models withdifferent ordering structures order coordination order consoli-dation andmulti-tier ordering hierarchyrdquo International Journalof Production Economics vol 72 no 3 pp 203ndash225 2001

[9] Y Duan J Luo and J Huo ldquoBuyer-vendor inventory coordina-tionwith quantity discount incentive for fixed lifetime productrdquoInternational Journal of Production Economics vol 128 no 1 pp351ndash357 2010

[10] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[11] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[12] C X Wang and S Webster ldquoChannel coordination for asupply chain with a risk-neutral manufacturer and a loss-averseretailerrdquo Decision Sciences vol 38 no 3 pp 361ndash389 2007

[13] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[14] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[15] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[16] J Cole Boeingrsquos Surplus Lot Filling Up Seattle Times 1998[17] PH Ritchken andC S Tapiero ldquoContingent claims contracting

for purchasing decisions in inventorymanagementrdquoOperationsResearch vol 34 no 6 pp 864ndash870 1986

[18] A A Tsay ldquoThe quantity flexibility contract and supplier-cus-tomer incentivesrdquoManagement Science vol 45 no 10 pp 1339ndash1358 1999

[19] A Burnetas and P Ritchken ldquoOption pricing with downward-sloping demand curves the case of supply chain optionsrdquoMan-agement Science vol 51 no 4 pp 566ndash580 2005

[20] B Xu Y Jia and L Liu ldquoThe decision models and coordinationof supply chain with one manufacturers and two retailers basedon CVaR criterionrdquo Journal of Shandong University (NaturalScience) vol 48 no 7 pp 101ndash110 2013

[21] M E Schweitzer and G P Cachon ldquoDecision bias in the news-vendor problem with a known demand distribution experi-mental evidencerdquoManagement Science vol 46 no 3 pp 404ndash420 2000

[22] K Maccrimmon and D A Wehrung Taking Risks The Man-agement of Uncertainty Free Press New York NY USA 1986

[23] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[24] T H Ho and J Zhang ldquoDesigning pricing contracts for bound-edly rational customers does the framing of the fixed feematterrdquoManagement Science vol 54 no 4 pp 686ndash700 2008

[25] T Feng L R Keller and X Zheng ldquoDecision making inthe newsvendor problem a cross-national laboratory studyrdquoOmega vol 39 no 1 pp 41ndash50 2011

[26] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Production and OperationsManagement vol 13 no 2 pp 135ndash149 2004

[27] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[28] C A Ingene and M E Parry ldquoCoordination and manufacturerprofit maximization the multiple retailer channelrdquo Journal ofRetailing vol 71 no 2 pp 129ndash151 1995

[29] V Padmanabhan and I P L Png ldquoManufacturerrsquos returnpolicies and retail competitionrdquo Marketing Science vol 16 no1 pp 81ndash94 1997

[30] Z Yao Y Wu and K K Lai ldquoDemand uncertainty and man-ufacturer returns policies for style-good retailing competitionrdquoProduction Planning amp Control vol 16 no 7 pp 691ndash700 2005

[31] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[32] T Xiao and X Qi ldquoPrice competition cost and demand disrup-tions and coordination of a supply chain with onemanufacturerand two competing retailersrdquoOmega vol 36 no 5 pp 741ndash7532008

[33] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[34] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of BankingampFinance vol 26no 7 pp 1443ndash1471 2002

[35] L Yang M Xu G Yu and H Zhang ldquoSupply chain coordina-tion with CVaR criterionrdquo Asia-Pacific Journal of OperationalResearch vol 26 no 1 pp 135ndash160 2009

[36] X Chen S Shum andD Simchi-Levi ldquoStable and coordinatingcontracts for a supply chain withmultiple risk-averse suppliersrdquoProduction and Operations Management vol 23 no 3 pp 379ndash392 2014

[37] MWu S X Zhu and R H Teunter ldquoA risk-averse competitivenewsvendor problem under the CVaR criterionrdquo InternationalJournal of Production Economics vol 156 pp 13ndash23 2014

[38] C-C Hsieh and Y-T Lu ldquoManufacturerrsquos return policy in atwo-stage supply chain with two risk-averse retailers andrandom demandrdquo European Journal of Operational Researchvol 207 no 1 pp 514ndash523 2010

[39] Z Liu L Chen and X Zhai ldquoSupply chain coordination basedon option contract and risk-averse retailersrdquo Systems Engineer-ing vol 31 no 9 pp 63ndash67 2013

[40] N Matsubayashi and Y Yamada ldquoA note on price and qualitycompetition between asymmetric firmsrdquo European Journal ofOperational Research vol 187 no 2 pp 571ndash581 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of